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ANALYTIC   GEOMETRY 


1 

A  SERIES  OF  MATHEMATICAL  TEXTS 

EDITED    BY 

EARLE   RAYMOND   HEDRICK 


THE   CALCULUS 

By    Ellery    Williams    Davis    and    William    Charles 
Brenke. 

PLANE   AND   SOLID  ANALYTIC   GEOMETRY 

By  Alexander  Ziwet  and  Louis  Allen  Hopkins. 

PLANE     AND     SPHERICAL     TRIGONOMETRY     WITH 
COMPLETE   TABLES 
By  Arthur  Monroe  Kenyon  and  Louis  Ingold, 

PLANE     AND     SPHERICAL     TRIGONOMETRY     WITH 
BRIEF   TABLES 
By  Arthur  Monroe  Kenyon  and  Louis  Ingold. 

THE   MACMILLAN  TABLES 

Prepared  under  the  direction  of  Earle  Raymond  Hedrick. 

PLANE   GEOMETRY 

By  Walter  Burton  Eord  and  Charles  Ammerman. 

PLANE  AND   SOLID  GEOMETRY 

By  Walter  Burton  Ford  and  Charles  Ammerman. 

SOLID   GEOMETRY 

By  Walter  Burton  Ford  and  Charles  Ammerman. 


ANALYTIC    GEOMETRY 


AND 


PRINCIPLES  OF  ALGEBRA 


BY 

ALEXANDER   ZIWET 

PROFESSOR   OF   MATHEMATICS,    THE    UNIVERSITY    OF    MICHIGAN 
AND 

LOUIS   ALLEN   HOPKINS 

INSTRUCTOR    IN    MATHEMATICS,    THE    UNIVERSITY   OF   MICHIGAN 


'Nzta  gorft 

THE   MACMILLAN   COMPANY 

1913 

All  rights  reserved 


COPTBIGHT,    1913, 

By  the  MACMILLAN  COMPANY. 


Set  up  and  electrotyped.     Published  November,  1913 


NortooolJ  ^ttw 

J.  8.  Cashing  Co.  —  Berwick  &  Smith  Co. 

Norwood,  Masa.,  U.S.A. 


QA' 


PREFACE 

The  present  work  combines  with  analytic  geometry  a  num- 
ber of  topics  traditionally  treated  in  college  algebra  that 
depend  upon  or  are  closely  associated  with  geometric  repre- 
sentation. Through  this  combination  it  becomes  possible  to 
show  the  student  more  directly  the  meaning  and  the  useful- 
ness of  these  subjects. 

The  idea  of  coordinates  is  so  simple  that  it  might  (and  per- 
haps should)  be  explained  at  the  very  beginning  of  the  study 
of  algebra  and  geometry.  Keal  analytic  geometry,  however, 
begins  only  when  the  equation  in  two  variables  is  interpreted 
as  defining  a  locus.  This  idea  must  be  introduced  very  gradu- 
ally, as  it  is  difficult  for  the  beginner  to  grasp.  The  familiar 
loci,  straight  line  and  circle,  are  therefore  treated  at  great 
length. 

Simultaneous  linear  equations  present  themselves  naturally 
in  connection  with  the  intersection  of  straight  lines  and  lead 
to  an  early  introduction  of  determinants,  whose  broad  useful- 
ness is  most  apparent  in  analytic  geometry. 

The  study  of  the  circle  calls  for  a  discussion  of  quadratic 
equations  which  again  leads  to  complex  numbers.  The  geo- 
metric representation  of  complex  numbers  will  present  no 
great  difficulty  because  the  student  is  now  somewhat  familiar 
with  the  idea  of  variables,  of  coordinates,  and  even  vectors 
(in  a  plane). 

The  discussion  of  the  conic  sections  is  preceded  by  the 
study,  especially  the  plotting,  of  curves  of  the  form  y  =  f{x), 


vi  PREFACE 

where  f(x)  is  a  polynomial  of  the  second,  third,  etc.  degree. 
In  connection  with  this  the  solution  of  numerical  algebraic 
equations  can  be  given  a  geometric  setting. 

In  the  chapters  on  the  conic  sections  only  the  most  essential 
properties  of  these  curves  are  given  in  the  text;  thus,  poles 
and  polars  are  discussed  only  in  connection  with  the  circle. 

Great  care  has  been  taken  in  presenting  the  fundamental 
problem  of  finding  the  slope  of  a  curve.  It  seemed  desirable 
and  quite  feasible  to  introduce  the  idea  of  the  derivative  (of 
a  polynomial  only)  in  connection  with  the  discussion  of  alge- 
braic equations.  The  calculus  method  of  finding  the  slope  of 
a  conic  section  has  therefore  been  explained,  in  addition  to 
the  direct  geometric  method. 

The  treatment  of  solid  analytic  geometry  follows  more  the 
usual  lines.  But,  in  view  of  the  application  to  mechanics, 
the  idea  of  the  vector  is  given  some,  prominence;  and  the 
representation  of  a  function  of  two  variables  by  contour  lines 
as  well  as  by  a  surface  in  space  is  explained  and  illustrated 
by  practical  examples. 

The  exercises  have  been  selected  with  great  care  in  order 
not  only  to  furnish  sufficient  material  for  practice  in  algebraic 
work  but  also  to  stimulate  independent  thinking  and  to  point 
out  the  applications  of  the  theory  to  concrete  problems.  The 
number  of  exercises  is  sufficient  to  allow  the  instructor  to 
make  a  choice. 

To  reduce  the  course  presented  in  this  book  to  about  one 
half  its  extent,  the  parts  of  the  text  in  small  type,  the  chap- 
ters on  solid  analytic  geometry,  and  the  more  difficult  prob- 
lems throughout  may  be  omitted. 

ALEXANDER   ZIWET, 

L.  A.  HOPKINS, 

E.  R.  HEDRICK,  Editor. 


CONTENTS 
PLANE  ANALYTIC   GEOMETRY 


PAGES 


Chapter  I.    Coordinates .  1-22 

Chapter  II.    The  Straight  Line 23-38 

Chapter  III.    Simultaneous  Linear  Equations  —  Determi- 
nants   ' .        .        .  39-57 

Part  I.       Equations    in    Two    Unknowns  —  Determi- 
nants of  Second  Order       ....  39-45 
Part  II.     Equations  in  Three  Unknowns  —  Determi- 
nants of  Third  Order         ....  46-57 

Chapter  IV.    Relations  between  Two  or  More  Lines  .        .  58-69 

Chapter  V.    Permutations  and  Combinations  —  Determi- 
nants of  any  Order 70-86 

Chapter  VI.    The  Circle  —  Quadratic  Equations  .  87-109 

Chapter  VII.    Complex  Numbers 110-130 

PartL       The  Various  Kinds  of  Numbers     .         .         .  110-116 
Part  II.     Geometric  Interpretation  of  Complex  Num- 
bers              .         .  117-130 

Chapter  VIII.    Polynomials  —  Numerical  Equations  .        .  131-168 

PartL       Quadratic  Function — Parabola      .         .         .  131-142 

Part  IL     Cubic  Function        .         .         .        .         .         .  143-147 

Part  III.   The  General  Polynomial         ....  148-157 

Part  IV.    Numerical  Equations 158-168 

Chapter  IX.    The  Parabola 169-197 

Chapter  X.    Ellipse  and  Hyperbola      .        .        .        .  198-222 

vii 


viii  CONTENTS 


PAGES 


Chapter  XI.    Conic  Sections  —  Equation  of  Second  Degree  223-247 

Part  I.       Definition  and  Classification  ....  223-231 

Part  II.     Reduction  of  General  Equation      .         .         .  232-247 

Chapter  XII.    Higher  Plane  Curves 248-276 

Part  I.       Algebraic  Curves 248-253 

Part  II.     Special   Curves  —  Defined   Geometrically  or 

Kineniatically 254-260 

Part  III.   Special  Transcendental  Curves       .         .         .  261-265 

Part  IV.   Empirical  Equations 266-276 

SOLID   ANALYTIC   GEOMETRY 

Chapter  XIII.    Coordinates 277-291 

Chapter  XIV.    The  Plane  and  the  Straight  Line         .        .  292-316 

Part  I.       The  Plane 292-306 

Part  II.     The  Straight  Line 307-316 

Chapter  XV.    The  Sphere 317-331 

Chapter  XV  I.    QUadric  Surf  aces  —  Other  Surf  aces      .        .  332-355 

Appendix  —  Note  on  Numerical  Multiplication  and  Division  356-367 

Answers 359-364 

Index 365-369 


ANALYTIC   GEOMETRY 


PLANE  ANALYTIC  aEOMETRY 

CHAPTER   I 
COORDINATES 

1.  Location  of  a  Point  on  a  Line.  The  position  of  a  point 
P  (Fig.  1)  on  a  line  is  fully  determined  by  its  distance  OP 
from  a  fixed  point  0  on  the  line,  if  we  know  on  which  side  of 
O  the  point  P  is  situated  (to  the  right  or  to  the  left  of  0  in 
Fig.  1).     Let  us  agree,  for  instance,  to  count  distances  to  the 


f 


Fig.  1 


right  of  0  as  positive,  and  distances  to  the  left  of  0  as  negative ; 
this  is  indicated  in  Fig.  1  by  the  arrowhead  which  marks  the 
positive  sense  of  the  line. 

The  fixed  point  0  is  called  the  origin.  The  distance  OP, 
taken  with  the  sign  +  if  P  lies,  let  us  say,  on  the  right,  and 
with  the  sign  —  when  P  lies  on  the  opposite  side,  is  called 
the  abscissa  of  P. 

It  is  assumed  that  the  unit  in  which  the  distances  are 
measured  (inches,  feet,  miles,  etc.)  is  known.  On  a  geographi- 
cal map,  or  on  a  plan  of  a  lot  or  building,  this  unit  is  indicated 
by  the  scale.  In  Fig.  1,  the  unit  of  measure  is  one  inch,  the 
abscissa  of  P  is  +2,  that  of  Q  is  —  1,  that  of  P  is  —  1/3. 

B  1 


2  PLANE  ANALYTIC  GEOMETRY  [I,  §  2 

2.  Determination  of  a  Point  by  its  Abscissa.  Let  us  select, 
on  a  given  line,  an  arbitrary  origin  0,  a  unit  of  measure,  and  a 
definite  sense  as  positive.  Then  any  real  number,  such  as  5, 
—  3,  7.35,  —  V2,  regarded  as  the  abscissa  of  a  point  F,  fully 
determines  the  position  of  P  on  the  line.  Conversely,  every 
point  on  the  line  has  one  and  only  one  abscissa. 

The  abscissa  of  a  point  is  usually  denoted  by  the  letter  x, 
which,  in  analytic  geometry  as  in  algebra,  may  represent  any 
real  or  complex  number. 

To  represent  a  real  point  the  abscissa  must  be  a  real  number. 
If  in  any  problem  the  abscissa  a;  of  a  point  is  not  a  real  num- 
ber, there  exists  no  real  point  satisfying  the  conditions  of  the 
problem. 

EXERCISES 

1.  "What  is  the  abscissa  of  the  origin  ? 

2.  With  the  inch  as  unit  of  length,  mark  on  a  line  the  points  whose 
abscissas  are  :  3,  —2,  VS,  —  1.25,  —  V5,  |,  —  i 

3.  On  a  railroad  line  running  east  and  west,  if  the  station  B  is  20  miles 
east  of  the  station  A  and  the  station  C  is  33  miles  east  of  A,  what  are  the 
abscissas  of  A  and  C  for  B  as  origin,  the  sense  eastward  being  taken  as 
positive  ? 

4.  On  a  Fahrenheit  thermometer,  what  is  the  positive  sense  ?  What 
is  the  unit  of  measure  ?  What  is  the  meaning  of  the  reading  66°  ? 
What  is  meant  by  —  7°  ? 

5.  A  water  gauge  is  a  vertical  post  carrying  a  scale  ;  the  mean  water 
level  is  generally  taken  as  origin.  If  the  water  stands  at  -|-  7  on  one  day 
and  at  —11  the  next  day,  the  unit  being  the  inch,  how  much  has  the 
water  fallen  ? 

6.  If  xu  X2  (read :  x  one,  x  two)  are  the  abscissas  of  any  two  points 
Pi,  P2  on  a  given  line,  show  that  the  abscissa  of  the  midpoint  between 
Pi  and  P2  is  ^  (xi  +  3^2)  •  Consider  separately  the  cases  when  Pi,  P2  lie 
on  the  same  side  of  the  origin  0  and  when  they  lie  on  opposite  sides. 


I,  §  3]  COORDINATES  3 

3.  Ratio  of  Division.  A  segment  AB  (Fig.  2)  of  a  straight 
line  being  given,  it  is  shown  in  elementary  geometry  how  to 
find  the  point  C  that  divides 
AB  in  a  given  ratio  k.  Thus, 
if  it  =  I,  the  point  G  such  that 

AC^2 
AB     5 

is   found  as  follows.     On  any 

line  through  A  lay  off  AD  =  2  and  AE  =  5 ;  join  B  and  E. 
Then  the  parallel  to  BE  through  D  meets  AB  at  the  required 
point  C. 

Analytically,  the  problem  of  dividing  a  line  in  a  given  ratio 
is  solved  as  follows.  On  the  line  AB  (Fig.  3)  we  choose  a 
point  0  as  origin  and  assign  a  positive  sense.  Then  the 
abscissas  Xj  of  A  and  X2  of  B  are  known.     To  find  a  point  G 

r — 1 


^:g:":Z->  ' 


Fig.  3 

which  divides  AB  in  the  ratio  of  division  k  =  AG/AB,  let  us 
denote  the  unknown  abscissa  of  G  by  x.     Then  we  have 

AG=x  —  Xi,   AB  =  X2  —  Xi', 

hence  the  abscissa  x  oi  G  must  satisfy  the  condition 

H/2  —  iCj 

whence 

yj  ^^  yJj  ~j~  /t  ( «^2  """  *yiy  } 

or,  if  we  write  Ax  (read :  delta  x)  for  the  "  difference  of  the 
a^s,"  I.e.  Ax  =  X2  —  Xi, 

x  =  Xi-\-k  '  Ax. 

Thus,  if  the  abscissas  of  A  and  B  are  2  and  7,  the  abscissas 


4         PLANE  ANALYTIC  GEOMETRY      [I,  §  3 

of  the  points  that  divide  AB  in  the  ratios  |,  i,  |,  |  are  3,  4^, 
8,  9^,  respectively.  Check  these  results  by  geometric  con- 
struction. 

If  the  segments  AC  and  AB  have  the  same  sense,  the  divi- 
sion ratio  k  is  positive.  For  example,  in  Fig.  3,  the  point  O 
lies  between  A  and  B ;  hence  the  division  ratio  fc  is  a  positive 
proper  fraction.  If  the  division  ratio  k  is  negative,  the  seg- 
ments AC  and  AB  must  have  opposite  sense,  so  that  B  and  C 
lie  on  the  opposite  sides  of  A. 

If  the  abscissas  of  A  and  B  are  again  2  and  7,  the  abscissa 
xof  C  when  A;  =  2,  -  1,  -  f,  -  .2  will  be  12,  -  3,  0,  1,  respec- 
tively. Illustrate  this  by  a  figure,  and  check  by  the  geometric 
construction. 

4.  Location  of  a  Point  in  a  Plane.  To  locate  a  point  in 
a  plane,  that  is,  to  determine  its  position  in  a  plane,  we  may 
proceed  as  follows.  Draw  two  lines  at  right  angles  in  the 
plane ;  on  each  of  these  take  the  point  of  intersection  O  as 
origin,  and  assign  a  definite  positive  sense  to  each  line,  e.g.  by 
marking  each  line  with  an  arrowhead.  It  is  usual  to  mark 
the  positive  sense  of  one  line  by  affixing  the  letter  x  to  it,  and 
the  positive  sense  of  the  other  line  by 
affixing  the  letter  ?/  to  it,  as  in  Fig.  4. 
These  two  lines  are  then  called  the  axes 
of  coordinates,  or  simply  the  axes.  We 
distinguish  them  by  calling  the  line  Ox  the 
a>axis,  or  axis  of  abscissas,  and  the  line  Oy 
the  ?/-axis,  or  axis  of  ordinates.  Now  project  the  point  P  on 
each  axis,  i.e.  let  fall  the  perpendiculars  PQ,  PR  from  P  on 
the  axes.  The  point  Q  has  the  abscissa  OQ  =  x  on  the  axis  Ox. 
The  point  R  has  the  abscissa  OR  =  y  on  the  axis  Oy.  The 
distance    OQ  =  RP=x   is    called    the    abscissa    of    P,    and 


y 

B 

--,P 

r 

j 

y\ 

X 

1 

JC 

~~D 

Q 

Fig. 

I 

I,  §6j 


COORDINATES 


y 

n  P'r— 

/ 

1 
1 

^ 

1 

!     X 

I      '           ^ 

1 
— jp- 

m  \ 

JY 

p"' 

OR  =  QP  =  2/  is  called  the  ordinate  of  P.  The  position  of  the 
point  P  in  the  plane  is  fully  determined  if  its  abscissa  x  and 
its  ordinate  y  are  both  given.  The  two  numbers  x,  y  are  also 
called  the  coordinates  of  the  point  P. 

5.  Signs  of  the  Coordinates.  Quadrants.  It  is  clear 
from  Fig.  4  that  x  and  y  are  the  perpendicular  distances  of  the 
point  P  from  the  two  axes.  It  should  be  observed  that  each 
of  these  numbers  may  be  positive  or 
negative,  as  in  §  1. 

The  two  axes  divide  the  plane  into 
four  compartments  distinguished  as  in 
trigonometry  as  the  first,  second,  third, 
and  fourth  quadrants  (Fig.  5).  It  is 
readily  seen  that  any  point  in  the  first 
quadrant  has  both  its  coordinates  posi- 
tive. What  are  the  signs  of  the  coordi- 
nates in  the  other  quadrants  ?  What  are  the  coordinates  of  the 
origin  0  ?  What  are  the  coordinates  of  a  point  on  one  of  the 
axes  ?  It  is  customary  to  name  the  abscissa  first  and  then 
the  ordinate ;  thus  the  point  (—3,  5)  means  the  point  whose 
abscissa  is  —  3  and  whose  ordinate  is  5. 

Every  point  in  the  plane  has  two  definite  real  numbers  as  co- 
ordinates; conversely,  to  every  pair  of  real  numbers  corresponds 
one  and  ordy  one  point  of  the  jiilane. 

Locate  the  points:  (6,  -2),  (0,  7),  (2-V3,  f),  (-4,  2V2), 
(-5,0). 

6.  Units.  It  may  sometimes  be  convenient  to  choose  the 
unit  of  measure  for  the  abscissa  of  a  point  different  from  the 
unit  of  measure  for  the  ordinate.  Thus,  if  the  same  unit,  say 
one  inch,  were  taken  for  abscissa  and  ordinate,  the  point  (3,  48) 
might  fall  beyond  the  limits  of  the  paper.     To  avoid  this  we 


6 


PLANE  ANALYTIC  GEOMETRY 


[I,  §6 


may  lay  off  the  ordinate  on  a  scale  of  i  inch.  When  different 
units  are  used,  the  unit  used  on  each  axis  should  always  be 
indicated  in  the  drawing.  ^  When  nothing  is  said  to  the  con- 
trary, the  units  for  abscissas  and  ordinates  are  always  under- 
stood to  be  the  same. 

7.  Oblique  Axes.  The  position  of  a  point  in  a  plane  can 
also  be  determined  with  reference  to  two  axes  that  are  7iot  at 
right  angles ;  but  the  angle  <o  between  these 
axes  must  be  given  (Fig.  6).  The  abscissa 
and  the  ordinate  of  the  point  P  are  then         /  y/ 

0/a>         X         / 

the  segments   OQ  =  x,   OB  =  y  cut  off  on      /\  Jg 

the  axes  by  the  parallels  through  P  to  the 

axes.     If  o)  =  |^7r,  i.e.  if  the  axes   are   at 

right  angles,   we  have  the  case  of   rectangular    coordinates 

discussed  in  §§4,  5.     In  what  follows,  the  axes  are  always 

taken    at    right    angles    unless    the    contrary    is    definitely 

stated. 

8.  Distance  of  a  Point  from  the  Origin. 

For  the  distance  r  =  OP  (Fig.  7)  of  the  point 
P  from  the  origin  O  we  have  from  the  right- 
angled  triangle  OQP: 

Fig.  7 


p 


where  x,  y  are  the  coordinates  of  P. 

If  the  axes  are  oblique  (Fig.  8),  with  the  angle 
xOy  =  (a^  we  have,  from  the  triangle  OQP,  in 
which  the  angle  at  Q  is  equal  to  ir  —  w,*  by  the 
cosine  law  of  trigonometry, 


Fig.  8 


r  =  Vx2  -\-y2  —  2  xy  cos  (tt  —  w)  =  Vx^  +  y"-^  +  2  xy  cos  w. 


*  In  advanced  mathematics,  angles  are  generally  measured  in  radians,  the 
symbol  tt  denoting  an  angle  of  180^. 


I,  §  9]  COORDINATES  7 

Notice  that  these  formulas  hold  not  only  when  the  point  P 
lies  in  the  first  quadrant,  but  quite  generally  wherever  the 
point  P  may  be  situated.     Draw  the  figures  for  several  cases. 

9.  Distance  between  Two  Points.  By  Fig.  9,  the  distance 
d  =  PiP2  between  two  points  Pi{xi,  y^  and  ^2(^2?  2/2)  can  be 
found  if  the  coordinates  of  the  two  points 
are  given.  For  in  the  triangle  P1QP2  we  ^ 
have 

PiQ  =  X2-Xi,    QPs  =  2/2  -  2/1 ; 
hence  Fio.  9 

(1)  e«  =  V(i»2-a:^i)2  + (2/2-2/1)2. 

If  we  write  Ax  (§  3)  for  the  "  difference  of  the  aj's  "  and  Ay 
for  the  ^'  difference  of  the  ^'s  ",  i.e. 

Ax  =  X2  —  Xi     and     Ay  =  2/2  —  2/1  > 
the  formula  for  the  distance  has  the  simple  form 

(2)  rZ  =  V(Ai»)2-|-(Ai/)2; 

or,  in  words, 

The  distance  between  any  two  points  is  equal  to  the  square  root 
of  the  sum  of  the  squares  of  the  differences  between  their  corre- 
sponding coordinates. 

Draw  the  figure  showing  the  distance  between  two  points 
(like  Fig.  9)  for  various  positions  of  these  points  and  show 
that  the  expression  for  d  holds  in  all  cases. 

Show  that  the  distance  between  two  points  Pi  {xi,  ?/i),  P2  (0:2, 2/2)  when 
the  axes  are  oblique,  with  angle  w,  is 

d  =  V{x2  -  xi)2+  (2/2  -  yi)'^  +  2(X2  -  xi) (2/2  -  y\)  cos  w 
=  \/(  Ax)2  +  (Ay)2  +  2  Ax  .  Ay  .  cos  w. 


8 


PLANE  ANALYTIC  GEOMETRY 


[I,  §  10 


10.  Ratio  of  Division.  If  two  points  P^  {x^ ,  2/1)  «**f^  ^2  fe  2/2) 
are  given  by  their  coordinates,  the  coordinates  x,  y  of  any  point 
Pon  the  line  P1P2  can  be  found  if  the  division  ratio  P^P/P^P^  =  k 
is  known  in  lohich  the  point  P  divides  the  segment  P^P^.  Let  Q^ , 
Q2,  Q  (Fig.  10),  be  the  projections  oi  P^,  P2,  P  on  the  axis  Ox ; 
then  the  point  Q  divides  Q1Q2  in  the  same  ratio  k  in  which 
P  divides  PiP^-  Now  as  OQi  =  aji, 
0^2=  ^2)  OQ  =  X,  it  follows  from  §  3 
that 

X=Xi  -\-k(X2—  Xi). 

In  the  same  way  we  find  by  projecting 
Pi,  P2f  P  on  the  axis  Oy  that 

Fig.  10 

2/  =  2/i  ^-^•0/2-2/l)• 
Thus,  the  coordinates  x,  y  oi  P are  found  expressed  in  terms 
of  the  coordinates  of  P^ ,  Po  and  the  division  ratio  k.      Putting 
again  X2  —  Xi  =  Ax,  2/2  —  2/1  =  ^2/ >  we  may  also  write 

x  =  Xi-}-k'  Ax,     y  =  yi-\-k'Ay. 

Here  again  the  student  should  convince  himself  that  the 
formulas  hold  generally  for  any  position  of  the  two  points,  by 
selecting  numerous  examples.  He  should  also  prove,  from  a 
figure,  that  the  same  expressions  for  the  coordinates  of  the 
point  P  hold  for  oblique  coordinates. 

As  in  §  3,  if  the  division  ratio  k  is  negative,  the  two 
segments  P1P2  and  P^P  must  have  opposite  sense,  so  that 
the  points  P  and  Pg  must  lie  on  opposite  sides  of  the 
point  Pi. 

Find,  e.g.,  the  coordinates  of  the  points  that  divide  the  seg- 
ment joining  (—  4,  3)  to  (6,  —  5)  in  the  division  ratios  k  =  ^, 
k  =  2,  fc=— 1,  k  =  —  1,  and  indicate  the  four  points  in  a 
figure. 


I J  11]  •         COORDINATES  9 

11.  Midpoint  of  a  Segment.  The  midpoint  P  of  a  segment 
P1P2  has  for  its  coordinates  the  arithmetic  means  of  the  corre- 
sponding coordinates  of  P^  and  P^ ;  that  is,  if  x-^ ,  2/1  are  the  co- 
ordinates of  Pi,  0-2,  2/2  those  of  P2,  the  division  ratio  being 
A;  =  I",  the  coordinates  of  the  midpoint  P  are  (§  10) 

a;  =  a?!  -j-  "2"  (^2      ^1)  =  2  (^1  1  •^2)5 
2/ =  ^1  +  i (2/2  -  2/1)  =  i (2/1  +  2/2). 

EXERCISES 

1.  With  reference  to  the  same  set  of  axes,  locate  the  points  (6,  4), 
(2,  -  i).  (-  6.4,  -  3.2),  (-4,  0),  (-  1,  5),  (.001,  -  4.01). 

2.  Locate  the  points  (-3,4),  (0,-1),  (6,  -  V2),  (1,-10^), 
(0,a),  (a,  6),  (3,  -2),  (-2,  v^). 

3.  If  a  and  6  are  positive  numbers,  in  what  quadrants  do  the  follow- 
ing points  lie  :  (a,  —  6),  (6,  a),  (a,  a),  (—  &,  &),  (—  &,  —  a)? 

4.  Show  that  the  points  (a,  6)  and  (a,  —  6)  are  symmetric  with 
respect  to  the  axis  Ox  ;  that  (a,  6)  and  (—a,  6)  are  symmetric  with  re- 
spect to  the  axis  Oy  ;  that  (a,  6)  and  (—  a,  —  6)  are  symmetric  with 
respect  to  the  origin. 

5.  In  the  city  of  Washington  the  lettered  streets  (A  street,  B  street, 
etc.)  run  east  and  west,  the  numbered  streets  (1st  street,  2d  street,  etc.) 
north  and  south,  the  Capitol  being  the  origin  of  coordinates.  The  axes 
of  coordinates  are  called  aivenues  ;  thus,  e.gr.,  1st  street  north  runs  one 
block  north  of  the  Capitol.  If  the  length  of  a  block  were  1/10  mile,  what 
would  be  the  distance  from  the  corner  of  South  C  street  and  East  5th 
street  to  the  corner  of  North  Q  street  and  West  14^h  street  ? 

6.  Prove  that  the  points  (6,  2),  (0,  -  6),  (7,  1)  lie  on  a  circle  whose 
center  is  (3,  —  2). 

7.  A  square  of  side  s  has  its  center  at  the  origin  and  diagonals  coin- 
cident with  the  axes  ;  what  are  the  coordinates  of  the  vertices  ?  of  the 
midpoints  of  the  s.ides  ?  ... 

8.  If  a  point  moves  jjarallel  to  the  axis  Oy,  which  of  its  coordinates 
remains  constant  ? 


10        PLANE  ANALYTIC  GEOMETRY     [I,  §  U 

9.   In  what  quadrants  can  a  point  lie  if  its  abscissa  is  negative  ?  its 
ordinate  positive  ? 

10.  Find  the  coordinates  of  the  points  which  trisect  the  distance  be- 
tween the  points  (1,  —  2)  and  (—  3,  4). 

11.  To  what  point  must  the  hne  segment  drawn  from  (2,  —3)  to 
(—3,  5)  be  extended  so  that  its  length  is  doubled  ?  trebled  ? 

12.  The  abscissa  of  a  point  is  —  3,  its  distance  from  the  origin  is  5 ; 
what  is  its  ordinate  ? 

13.  A  rectangular  house  is  to  be  built  on  a  corner  lot,  the  front,  30  ft. 
wide,  cutting  off  equal  segments  on  the  adjoining  streets.  If  the  house  is 
20  ft.  deep,  find  the  coordinates  (with  respect  to  the  adjoining  streets)  of 
the  back  corners  of  the  house. 

14.  A  baseball  diamond  is  90  ft.  square  and  pitcher's  plate  is  60  ft. 
from  home  plate.  Using  the  foul  lines  as  axes,  find  the  coordinates  of 
the  following  positions  : 

(a)  pitcher's  plate  ; 

(6)  catcher  8  ft.  back  of  home  plate  and  in  line  with  second  base  ; 

(c)  base  runner  playing  12  ft.  from  first  base  ; 

(d)  third  baseman  playing  midway  between  pitcher's  plate  and  third 
base  (before  a  bunt) ; 

(e)  right  fielder  playing  90  ft.  from  first  and  second  base  each. 

16.  How  far  does  the  ball  ^o  in  Ex.  14  if  thrown  by  third  baseman 
in  position  (d)  to  second  base  ? 

16.  If  right  fielder  (Ex.  14)  catches  a  ball  in  position  (e)  and  throws 
it  to  third  base  for  a  double  play,  how  far  does  the  ball  go  ? 

17.  A  park  600  ft.  long  and  400  ft.  wide  has  six  lights  arranged  in  a 
circle  about  a  central  light  cluster.  All  the  lights  are  200  ft.  apart,  and 
the  central  cluster  and  two  others  are  in  a  line  parallel  to  the  length  of 
the  park.  What  are  the  coordinates  of  all  the  lights  with  respect  to  two 
boundary  hedges  ? 

18.  With  respect  to  adjoining  walks,  three  trees  have  coordinates 
(30  ft.,  8  ft.),  (20  ft.,  45  ft.),  (-  27  ft.,  14  ft.),  respectively.  A  tree  is  to 
be  planted  to  form  the  fourth  vertex  of  a  parallelogram;  where  should  it 
be  placed  ?     (Three  possible  positions ;  best  found  by  division  ratio.) 


I,  §  12] 


COORDINATES 


11 


Fig.  11 


12.   Area  of  a  Triangle  with  One  Vertex  at  the  Origin. 

Let  one  vertex  of  a  triangle  be  the  origin,  and  let  the  other 
vertices  be  P^  {x^,  2/1)  and  P^  (x^,  y^.  Draw  through  P^  and 
P2  lines  parallel  to  the  axes  (Fig.  11).  The 
area  A  of  the  triangle  is  then  obtained  by 
subtracting  from  the  area  of  the  circum- 
scribed rectangle  the  areas  of  the  three  non- 
shaded  triangles ;  i.e. 
A  =  x{y^  -  i  a;i?/i  -\x^^-\  (x^  -x^  {y^  -  y^) 

=  i{^iy2  -  a^22/i). 
This  formula  gives  the  area  with  the  sign  -|-  or  —  according 
as  the  sense  of  the  motion  around  the  perimeter   OP1P2O  is 
counterclockwise  (opposite  to  the  rotation  of  the  hands  of  a 
clock)  or  clockwise. 

For  numerical  computation  it  is  most  convenient  to  write 
down  the  coordinates  of  the  two  points  thus : 

^1  2/1 

«2  2/2 
and  to  take  half  the  difference  of  the  crosswise  products.    The 
formula  is  therefore  often  written  in  the  form 


=i 

X, 
X, 

2/2 

^1  Vi 

X 

I  2/2 

where  the  symbol 


stands  for  x^yc^—x-^i,  and  is  called  a  determinant  (of  the  second 
order). 

Thus,  the  area  of  the  triangle  formed  by  the  origin  with  the 
pair  of  points  (4,  3)  and  (2,  5)  is 

.     4  3 


2  5 


=:i(4x5-2x3)  =  7. 


12 


PLANE  ANALYTIC  GEOMETRY 


[I,  §  13 


13.  Translation  of  Axes.  Instead  of  the  origin  0  and  the 
axes  Ox,  Oy  (Fig.  12),  let  us  select  a  new  origin  0'  (read :  O 
prime)  and  new  axes  0'x\  O'y',  parallel  to  the  old  axes.  Then 
any  point  P  whose  coordinates  with  reference  to  the  old  axes 
are  OQ=:x,  QP  =  y  will  have  with 
reference  to  the  new  axes  the  coordi- 
nates 0'Q'  =  x',  Q'P=y'',  and  the 
figure  shows  that  if  7i,  k  are  the  co- 
ordinates of  the  new  origin,  then 


X  —  x'  +  h, 
y  =  y'-\-k. 


yl 

y 

1 

— i — j> 

1       y\ 

k 

h       \                  \         X 

0 

Q 

Fig.  V. 


The  change  from  one  set  of  axes  to  a  new  set  is  called  a 
transformation  of  coordinates.  In  the  present  case,  where  the 
new  axes  are  parallel  to  the  old,  this  transformation  can  be 
said  to  consist  in  a  translation  of  the  axes. 

14.  Area  of  Any  Triangle.  Let  Pi(aji,  y^,  P^ix^,  2/2)? 
P3  (ajg ,  2/3)  be  the  vertices  of  the  triangle  (Fig.  13).  If  we  take 
one  of  these  vertices,  say  P3,  as  new 
origin,  with  the  new  axes  parallel  to  the  ^ 
old,  the  new  coordinates  of  Pi ,  Pg  will  be : 


I 


Jb^  Jbo%      Jb 


Xo X'\ 


y'i=yi-ys,  y'2  =  y2-: 


Hence,  by  §  12,  the  area  of  the  triangle 
AAA  is 


Fig. 13 


A  =  i {x'yy'.,-x'^\)  =  l  [{xi  -  xs)  (2/2  -  2/3) -  {xo  -  x^)  (y,  -  y.,)] 

For  numerical  computation  it  is  best  to  put  down  the  coordi- 
nates of  the  three  points  with  a  1  after  each  pair,  thus : 


I,  §  14] 


COORDINATES 


13 


flJl 

2/1 

1 

x^ 

2/2 

1 

% 

2/.3 

1 

Then  add  the  three  products  formed  by  following  the  full  lines 
and  subtract  the  three  i^roducts  formed  by  following  the  dotted 
lines  as  indicated  in  the  accompany- 
ing scheme,  i.e.  form  the  determinant 
(of  the  third  order) 

=  a^?/2  +  a^22/3  +  ^32/i  -  «^32/2  -  ^^1  -  aJi2/3.    / 

This  is   equal  to  the  expression  in    \ 
the  square  brackets  above,  i.e.  to  2  A. 
Therefore 


^=h 


Here  as  in  §  12  the  sign  of  the  area  is  +  or  —  according  as 
the  sense  of  the  motion  along  the  perimeter  P^PoP^P^  is  coun- 
terclockwise or  clockwise. 


a^ 

2/1 

1 

x^ 

2/2 

1 

Xz 

2/3 

1 

EXERCISES 

1.  Find  the  areas  of  the  triangles  having  the  following  vertices : 

(a)   (1,  3),  (5,  2),  (4,  6)  ;  (6)    (-2,  1),  (2,  -  3),  (0,  -  6)  ; 

(c)    (a,  6),  (a,  0),  (0,  h) ;  {d)  (4,  3),  (6,  -  2),  (-  1,  5). 

2.  Show  that  the  area  of  the  triangle  whose  vertices  are  (7,  —  8), 
(—  3,  2),  (—5,  —4)  is  four  times  the  area  of  the  triangle  formed  by- 
joining  the  midpoints  of  the  sides. 

3.  Find  the  area  of  the  quadrilateral  whose  vertices  are  (2,  3),  (—  1, 
-1),  (-4,2),  (-3,6). 

4.  Find  the  area  of  the  triangle  whose  vertices  are  (a,  0),  (0,  6), 
C-c,  -c). 

5.  Find  the  area  of  the  triangle  (1,  4),  (3,  -2),  (-3,  16).  What 
does  your  result  show  about  these  points  ? 


14 


PLANE  ANALYTIC    GEOMETRY 


[I,  §  14 


6.  Find  the  area  of  the  triangle  (a,  h  +  c),  (6,  c  +  a),  (c,  a  +  h). 
What  does  the  result  show  whatever  the  values  of  a^h^c'} 

7.  Show  that  the  points  (3,  7),  (7,  3),  (8,  8)  are  the  vertices  of  an 
isosceles  triangle.  What  is  its  area  ?  Show  that  the  same  is  true  for  the 
points  (a,  6),  (6,  a),  (c,  c),  whatever  a,  6,  c,  and  find  the  area, 

8.  Find  the  perimeter  of  the  triangle  whose  vertices  are  (3,  7),  (2, 
—  1),  (5,  3).     Is  the  triangle  scalene  ?     What  is  its  area  ? 

15.  Statistics.  Related  Quantities.  If  pairs  of  values 
of  two  related  quantities  are  given,  each  of  these  pairs  of 
Values  is  represented  by  a  point  in  the  plane  if  the  value  of 
one  quantity  is  represented  by  the  abscissa  and  that  of  the 
other  by  the  ordinate  of  the  point.  A  curved  line  joining 
these  points  gives  a  vivid  idea  of  the  way  in  which  the  two 
quantities  change.  Statistics  and  the  results  of  scientific  ex- 
periments are  often  represented  in  this  manner. 


EXERCISES 

1.   The  population  of  the  United  States,  as  shown  by  the  census  reports, 
is  approximately  as  given  in  the  following  table  : 


Tear 

1790 

1800 

'10 

'20 

'30 

'40 

'50 

'60 

'TO 

'80 

'90 

1900 

'10 

Millions 

4 

5 

7 

10 

13 

17 

23 

31 

39 

50 

63 

76 

92 

Mark  the  points  corresponding  to  the  pairs  of  numbers  (1790,  4), 
(1800,  5),  etc.,  on  squared  pager,  representing  the  time  on  the  horizontal 
axis  and  the  population  vertically.     Connect  these  points  by  a  curved  line. 

2.  From  the  figure  of  Ex.  1,  estimate  approximately  the  population 
of  the  United  States  in  1875  ;  in  1905  ;  in  1915. 

3.  From  the  figure  of  Ex.  1,  estimate  approximately  when  the  popula- 
tion was  25  millions  ;  60  millions  ;  when  it  will  be  100  milhons. 

4.  Draw  a  figure  to  represent  the  growth  of  the  population  of  yom' 
own  State,  from  the  figures  given  by  the  Census  Reports. 


I,  §  15] 


COORDINATES 


15 


[Other  data  suitable  for  statistical  graphs  can  be  found  in  large  quan- 
tity in  the  Census  Keports ;  in  the  Crop  Reports  of  the  government ;  in 
the  quotations  of  the  market  prices  of  food  and  of  stocks  and  bonds  ;  in 
the  World  Almanac  ;  and  in  many  other  books.  ] 

5.   The  temperatures  on  a  certain  day  varied  hour  by  hour  as  follows  : 


A.M. 

N. 

P.M. 

Time  .  . 
Temp.  .  . 

6 
50 

7 
52 

8 
55 

9 
60 

10 
64 

11 
67 

12 
70 

1 

72 

2 

74 

3 
75 

4 

74 

5 

72 

6 
69 

7 
65 

8 
60 

9 
57 

Draw  a  figure  to  represent  these  pairs  of  values. 

6.  In  experiments  on  stretching  an  iron  bar,  the  tension  t  (in  tons) 
and  the  elongation  E  (in  thousandths  of  an  inch)  were  found  to  be  as 
follows : 


t  (in  tons) 

JS  (in  thousandths  of  an  inch) 


6 
60 


8 
81 


10 
103 


Draw  a  figure  to  represent  these  pairs  of  values. 

[Other  data  can  be  found  in  books  on  Physics  and  Engineering.] 

7.  By  Hooke's  law,  the  elongation  ^  of  a  stretched  rod  is  supposed 
to  be  connected  with  the  tension  t  by  the  formula  E  =  c  -t,  where  c  is  a 
constant.  Show  that  if  c  =  10,  with  the  units  of  Ex.  6,  the  values  of  E 
and  t  would  be  nearly  the  same  as  those  of  Ex.  6.  Plot  the  values  given 
by  the  formula  and  compare  with  the  figure  of  Ex.  6. 

8.  The  distances  through  which  a  body  will  fall  from  rest  in  a  vacuum 
in  a  time  t  are  given  by  the  formula  s  =  16  t^,  approximately,  if  t  is  in 
seconds  and  s  is  in  feet.     Show  that  corresponding  values  of  s  and  t  are 


2 
64 


3 
144 


4 

256 


5 

400 


6 
576 


Draw  a  figure  to  represent  these  pairs  of  values. 


16  PLANE  ANALYTIC  GEOMETRY  [I,  §  16 

16.  Polar  Coordinates.  The  position  of  a  point  P  in  a 
plane  (Fig.  14)  can  also  be  assigned  by  its  distance  OP=r 
from  a  fixed  point,  or  pole,  0,  and  the  angle  xOP  =  (f>,  made 
by  the  line  OP  with  a  fixed  line  Ox,  the  polar  axis.  The  dis- 
tance r  is  called  the  radius  vector,  the  angle  <^  the  polar  angle 
(or  also  the  vectorial  angle,  azimuth,  qmpU-  j» 
tvAie^  or   anomaly),  of  the  point  P.     The            ^^^^ 

radius  vector  r  and  the  polar  angle  <^  are    O'^^^ £> 

called  the  polar  coordinates  of  P.  ^'^'  ^* 

Locate  the  points:  (5,  \it),  (6,  |7r),  (2,  140°),  (7,  307°), 
(V5,  tt),  (4,  0°). 

To  obtain  for  every  point  in  the  plane  a  single  definite  pair  of  polar 
coordinates  it  is  sufficient  to  take  the  radius  vector  r  always  positive  and 
to  regard  as  polar  angle  the  positive  angle  between  0  and  2  tt  (0  ^  0  <  2  tt) 
through  which  the  polar  axis  (regarded  as  a  half-line  or  ray  issuing  from 
the  pole  0)  must  be  turned  about  the  pole  O  in  the  counterclockwise  sense 
to  pass  through  P.  The  only  exception  is  the  pole  0  for  which  r  =  0, 
while  the  polar  angle  is  indeterminate. 

But  it  is  not  necessary  to  confine  the  radius  vector  to  positive  values 
and  the  polar  angle  to  values  between  0  and  2  7r.  A  single  definite  point 
P  will  correspond  to  every  pair  of  real  values  of  r  and  0,  if  we  agree  that 
a  negative  value  of  the  radius  vector  means  that  the  distance  r  is  to  be 
laid  off  in  the  negative  sense  on  the  polar  axis,  after  being  turned  through 
the  angle  0,  and  that  a  negative  value  of  <j>  means  that  the  polar  axis 
should  be  turned  in  the  clockwise  sense. 

The  polar  angle  is  then  not  changed  by  adding  to  it  any  positive  or 
negative  integral  multiple  of  2  tt  ;  and  a  point  whose  polar  coordinates  are 
r,  0  can  also  be  described  as  having  the  coordinates  —  r,  (p  ±  ir. 

Locate  the  points : 

(3,  -i^),  (a,  -Itt),  (-5,  75°),  (-3,  -20°). 

17.  Transformation  from  Cartesian  to  Polar  Coordinates, 

and  vice  versa.     The  coordinates   OQ  =  x,  QP=zy,  defined  in 
§  4,   are   called    cartesian    coordinates,   to    distinguish    them 


I,  §  18]  COORDINATES  17 

from  the  polar  coordinates.  The  term  is  derived  from  the 
Latin  form,  Cartesius,  of  the  name  of  Rene  Descartes,  who 
first  applied  the  method  of  coordinates  systematically  (1637), 
and  thus  became  the  founder  of  analytic  geometry. 

The  relation  between  the  cartesian  and  polar  coordinates  of 
one  and   the   same   point   P  appears   from 
Fig.  15.     We  have  evidently  : 


V 

^^V       X         \     X 

0 

Q 

Fig.  15 

x=:r  COS  <j>y  y—  ^^^  +  2/^ 

2/  =  rsin<^,     ^^  tan</,=:^. 


18.    Distance  between  Two  Points  in  Polar  Coordinates. 

If  two  points  Pi ,  Po  are  given  by  their  polar  coordinates,  r^ , 
<^i  and  r^ ,  <^2  ?  the  distance  d  =  PjPg  between 
them  is  found  from  the  triangle  OP1P.2  (Fig.  16), 
by  the  cosine  law  of  trigonometry,  if  we  ob- 
serve that  the  angle  at  O  is  equal  to  ±  (<^2— <^i)  • 


d  =  Vri^  -1-^2^  —  2  rirs  cos  (<^2  —  <^i)- 


Fig.  Iti 


EXERCISES 

1.  Find  the  distances  between  the  points :    (2,   |  tt)    and    (4,   |  tt)  ; 
(a,  Itt)  and  (3  a,  ^tt). 

2.  Find  the  cartesian  coordinates  of  the  points  (5,  ^tt),  (6,  — -^tt), 
(4a7r),  (2,  Itt),  (7,  7r),(6,  -tt),  (4,0),  (-3,60°),  (-5,  -90^^). 

3.  Find  the  polar  coordinates  of  the  points  (\/3, 1),  (—  V3,  1),  (1,  —1), 
i-h  -i)»  (-«'  «)• 

4.  Find  an  expression  for  the  area  of  the  triangle  whose  vertices  are 
(0,  0),  (n,  0i),  and  (ro ,  02). 

5.  Find  the  area  of  the  triangle  whose  vertices  are  (vi ,  e6i),  (r2 ,  02)? 

(»'3,  03). 

c 


18 


PLANE  ANALYTIC  GEOMETRY 


[I,  §  19 


6.  Find  the  radius  vector  of  the  point  P  on  the  Une  joining  the  points 
-Pi  {ii'i  1  0i)  and  P-z  (r2,  ^2)  sucli  that  the  polar  angle  of  P  is  ^(0i  +  02)  • 

7.  If  the  axes  are  oblique  with  angle  w,  what  arc  the  relations  existing 
between  the  cartesian  and  polar  coordinates  of  a  point  ? 

19.  Projection  of  Vectors.  A  straight  line  segment  AB 
of  definite  length,  direction,  and  sense  (indicated  by  an  arrow- 
head, pointing  from  A  to  B)  is  called  a  vector.  The  projection 
A'B'  (Figs.  17,  18)  of  a  vector  AB  on  an  axis,  i.e.  on  a  line  I 


on  which  a  definite  sense  has  been  selected  as  positive,  is  the 

product  of  the  length  (or  absolute  value)  of  the  vector  AB  into 

the  cosine  of  the  angle  between  the  positive  senses  of  the  axis  and 

the  vector  : 

A'B' =  AB  cos  a. 

The  positive  sense  of  the  axis  (drawn  through  the  initial  point 
of  the  vector)  makes  with  the  vector  two  angles  whose  sum  is 
2  IT  =  360°.      As    their  cosines 
are  the  same  it  makes  no  differ- 
ence which  of  the  two  angles  is 
used. 

With  these  conventions  it  is 
readily  seen  that  the  sum  of  the 
projections  of  the  sides  of  an 
open  polygon  on  any  axis  is  equal 
to  the  projection  of  the   closing 

side  on  the  same  axis,  the  sides   of  the   open  polygon  being 
taken  in  the  same  sense  around  the  perimeter.     Thus,  in  Fig.  19, 


Fig.  19 


I,  §  20]  COORDINATES  19 

the  vectors  P1P2)  A^s?  •••  AA  are  inclined  at  the  angles 
Ui,  02,  •"  a;i  to  the  axis  I;  the  closing  line  PiPq  makes  the 
angle  «with  ?;  its  projection  is  P'lP'e]   and  we  have 

P1P2  cos  «!  H-  P2P3  cos  ao  4-  P3P4  cos  ccg  +  P4P5  cos  a^  -\-  P^Pq  cos  «5 

=  P'iP'6  =  PiP6COsa. 

For,  if  the  abscissas  of  Pj,  Pg ,  •  •  •  Pe  measured  along  I,  from 
any  origin  0  on  /,  are  Xi,  X2,  •••  iCg ,  the  projections  of  the 
vectors  are  iCg  —  iCi ,  a^g  —  ajg ,  etc.,  so  that  our  equation  becomes 
the  identity : 

•^2  —  ^  ~r  ^3  —  ^2  ~r  *^4  —  -^s  "h  -^o  —  ^4  I   -^6  —  ^5^^  ^6  —  "^i* 

20.    Components  and  Resultants  of  Vectors.     In  physics, 
forces,  as  well  as  velocities,  accelerations,  etc.,  are  represented 
by  vectors  because  such  magnitudes  have  not  only  a  numerical 
value   but  also  a  definite    direction   and 
sense.  ? "^^v 

According  to  the  j^^^i^'^^^^^^ogram  law  of       Z^^-"  / 

physics,  two  forces  OPi,  OP2,  acting  on  ^  fig.  20  ' 
the  same  particle,  are  together  equivalent 
to  the  single  force  OP  (Fig.  20),  whose  vector  is  the  diagonal 
of  the  parallelogram  formed  with  OPi,  OP2  as  adjacent 
sides.  The  same  law  holds  for  simultaneous  velocities  and 
accelerations,  and  for  simultaneous  or  consecutive  rectilinear 
translations.  The  vector  OP  is  called  the  resultant  of  OP^ 
and  OP2 ,  and  the  vectors  OPi ,  OP2  are  called  the  components 
of  OP. 

To  construct  the  resultant  it  suffices  to  lay  off  from  the  ex- 
tremity of  the  vector  OPi  the  vector  P^P  =  OP2 ;  the  closing 
line  OP  is  the  resultant.     This  leads  at  once  to  finding  the 


20 


PLANE  ANALYTIC  GEOMETRY 


[I,  §  20 


resultant  OP  of  any  num- 
ber of  vectors,  by  adding 
the  component  vectors  geo- 
metrically, i.e.  putting  them 
together  endwise  succes- 
sively, as  in  Fig.  21,  where 
the  dotted  lines  need  not 
be  drawn. 

By    §19,   the   projection  ^'''- ^^ 

of  the  resultant  on  any  axis  is  equal  to  the  sum  of  the  pro- 
jections of  all  the  components  on  the  same  axis. 


EXERCISES 

1.  The  cartesian  coordinates  ic,  y  of  any  point  P  are  the  projections  of 
its  radius  vector  OP  on  the  axes  Ox,  Oy.     (See  §  16.) 

2.  The  projection  of  any  vector  AB  on  the  axis  Ox,  is  the  difference 
of  the  abscissas  of  A  and  B ;  similarly  for  Oy. 

3.  A  force  of  10  lb.  is  inclined  to  the  horizon  at  60° ;  find  its  hori- 
zontal and  vertical  components. 

4.  A  ship  sails  40  miles  N.  60°  E.,  then  24  miles  N.  45°  E.  How  far 
is  the  ship  then  from  its  starting  point  ?     How  far  east  ?     How  far  north  ? 

5.  A  point  moves  5  ft.  along  one  side  of  an  equilateral  triangle,  then 
6  ft.  parallel  to  the  second,  and  finally  8  ft.  parallel  to  the  third  side. 
What  is  the  distance  from  the  starting  point  ? 

6.  The  sum  of  the  projections  of  the  sides  of  any  closed  polygon  on 
any  axis  is  zero. 

7.  If  three  forces  acting  on  a  particle  are  parallel  and  proportional  to 
the  sides  of  a  triangle,  the  forces  are  in  equilibrium,  i.e.  their  resultant  is 
zero.     Similarly  for  any  closed  polygon. 

8.  Find  the  resultant  of  the  forces  OPi ,  OP2 ,  OP3 ,  OP4,  0P&,  if 
the  coordinates  of  Pi,  P2 ,  P3,  P4,  P5 ,  with  O  as  origin,  are  (3,  1), 
(1,  2),  (-1,  3),  (-2,  -2),  (2,  -2).  (Resolve  each  force  into  its 
components  along  the  axes.) 


I,  §  21]  COORDINATES  21 

9.  If  any  number  of  vectors  (in  the  same  plane) ,  applied  at  the  ori- 
gin, are  given  by  the  coordinates  x,  y  of  their  extremities,  the  length  of 
the  resultant  is  =V{IiX)'^  -\-{I>yy^  (where  2x  means  the  sum  of  the  ab- 
scissas, Sy  the  sum  of  the  ordinates) ,  and  its  direction  makes  with  Ox  an 
angle  a  such  that  tan  a  =  I,y/I>x. 

10.  Find  the  horizontal  and  vertical  components  of  the  velocity  of  a 
ball  when  moving  200  ft./sec.  at  an  angle  of  30°  to  the  horizon. 

11.  Six  forces  of  1,  2,  3,  4,  5,  6  lb.,  making  angles  of  60°  each  with 
the  next,  are  applied  at  the  same  point,  in  a  plane  ;  find  their  resultant. 

12.  A  particle  at  one  vertex  of  a  square  is  acted  upon  by  three  forces 
represented  by  the  vectors  from  the  particle  to  the  other  three  vertices  ; 
find  the  resultant. 

21.  Geometric  Propositions.  In  using  analytic  geometry 
to  prove  general  geometric  propositions,  it  is  generally  conven- 
ient to  select  as  origin  a  prominent  point  in  the  geometric 
figure,  and  as  axes  of  coordinates  prominent  lines  of  the  figure. 
But  sometimes  greater  symmetry  and  elegance  is  gained  by 
taking  the  coordinate  system  in  a  general  position.  (See,  e.g., 
Exs.  14,  17,  18,  below.) 

MISCELLANEOUS  EXERCISES 

1.  A  regular  hexagon  of  side  1  has  its  center  at  the  origin  and  one 
diagonal  coincident  with  the  axis  Ox  ;  find  the  coordinates  of  the  vertices. 

2.  Show  by  similar  triangles  that  the  points  (1,  4),  (3,  —  2),  (—  2, 
13)  lie  on  a  straight  line. 

3.  If  a  square,  with  each  side  5  units  in  length,  is  placed  with  one 
vertex  at  the  origin  and  a  diagonal  coincident  with  the  axis  Ox,  what  are 
the  coordinates  of  the  vertices  ? 

4.  If  a  rectangle,  with  two  sides  3  units  in  length  and  two  sides 
3  VS  units  in  length,  is  placed  with  one  vertex  at  the  origin  and  a  diagonal 
along  the  axis  Ox,  what  are  the  coordinates  of  the  vertices?  There  are  two 
possible  positions  of  the  rectangle  ;  give  the  answers  in  both  cases. 


22  PLANE  ANALYTIC  GEOMETRY  [I,  §  21 

6.  Show  that  the  pomts  (0,  -  1),  (-2,  3),  (6,  7),  (8,  3)  are  the 
vertices  of  a  parallelogram.    Prove  that  this  parallelogram  is  a  rectangle. 

6.  Show  that  the  points  (1,  1),  (-1,  —1),  (  +  V3,  —  >/3)  are  the 
vertices  of  an  equilateral  triangle. 

7.  Show  that  the  points  (6,  6),  (3/2,  -  3),  (-  3,  12),  (-  J^^,  3)  are 
the  vertices  of  a  parallelogram. 

8.  Find  the  radius  and  the  coordinates  of  the  center  of  the  circle  pass- 
ing through  the  three  points  (2,  3),  (-2,  7),  (0,  0). 

9.  The  vertices  of  a  triangle  are  (0,  6),  (4,  —3),  (—5,  6).  Find  the 
lengths  of  the  medians  and  the  coordinates  of  the  centroid  of  the  triangle, 
i.e.  of  the  intersection  of  the  medians. 

Prove  the  following  propositions  : 

10.  The  diagonals  of  any  rectangle  are  equal. 

11.  The  distance  between  the  midpoints  of  two  sides  of  any  triangle 
is  equal  to  half  the  third  side. 

12.  The  distance  between  the  midpoints  of  the  non-parallel  sides  of  a 
trapezoid  is  equal  to  half  the  sum  of  the  parallel  sides. 

13.  In  a  right  triangle,  the  distance  from  the  vertex  of  the  right  angle 
to  the  midpoint  of  the  hypotenuse  is  equal  to  half  the  hypotenuse. 

14.  The  line  segments  joining  the  midpoints  of  the  adjacent  sides  of  a 
quadrilateral  form  a  parallelogram. 

15.  If  two  medians  of  a  triangle  are  equal,  the  triangle  is  isosceles. 

16.  In  any  triangle  the  sum  of  the  squares  of  any  two  sides  is  equal 
to  twice  the  square  of  the  median  drawn  to  the  midpoint  of  the  third  side 
plus  half  the  square  of  the  third  side. 

17.  The  line  segments  joining  the  midpoints  of  the  opposite  sides  of 
any  quadrilateral  bisect  each  other. 

18.  The  sum  of  the  squares  of  the  sides  of  a  quadrilateral  is  equal  to 
the  sum  of  the  squares  of  the  diagonals  plus  four  times  the  square  of  the 
line  segment  joining  the  midpoints  of  the  diagonals. 

19.  The  difference  of  the  squares  of  any  two  sides  of  a  triangle  is  equal 
to  the  difference  of  the  squares  of  their  projections  on  the  third  side. 

20.  The  vertices  (xi,  y{),  (x^,  2/2),  (arg,  Vz)  of  a  triangle  being  given, 
find  the  centroid  (intersection  of  medians). 


Vj>A^ 


CHAPTER   II 

THE   STRAIGHT   LINE 

22.  Line  Parallel  to  an  Axis.  When  the  coordinates  x,  y 
of  a  point  P  with  reference  to  given  axes  Ox,  Oy  are  known, 
the  position  of  P  in  the  plane  of  the  axes  is  determined  com- 
pletely and  uniquely.  Suppose  now 
that  only  one  of  the  coordinates  is 
given,  say,  a?  =  3 ;  what  can  be  said 
about  the  position  of  the  point  P? 
It  evidently  lies  somewhere  on  the 
line  AB  (Fig.  22)  that  is  parallel  to 
the  axis  Oy  and  Jias  the  distance  3 
from  Oy.  Every  point  of  the  line  AB 
has  an  abscissa  x  =  3,  and  every  point 
whose  abscissa  is  3  lies  on  the  line  AB. 
say  that  the  equation  aj  =  3 


^rt^ 


A 
Fig.  22 


3   \4  \s 


For  this  reason  we 


represents  the  line  AB;  we  also  say  that  a;  =  3  is  the  equation 
of  the  line  AB. 

More  generally,  the  equation  x=a,  where  a  is  any  real 
number,  represents  that  parallel  to  the  axis  Oy  whose  distance 
from  Oy  is  a.  Similarly,  the  equation  y  =  h  represents  a 
parallel  to  the  axis  Ox. 

EXERCISES      ^*  \ 

Draw  the  lines  represented  by  the  equations  : 

1.  x=-2.  4.   5x  =  7.  7.   3a; +  1  =  0. 

2.  ic  =  0.  6.   y  =  0.  8.    10-3y  =  0. 

3.  a:  =  12.5,  6.   2y=-7.  9.   ?/=±2. 

23 


0 


i^. 


24  PLANE  ANALYTIC  GEOMETRY  [II,  §  23 

23.  Line  through  the  Origin.  Let  us  next  consider  any 
line  *  through  the  origin  0,  such  as  the  line  OP  in  Fig.  23. 
The  points  of  this  line  have  the  prop- 
erty that  the  ratio  y/x  of  their  coordi- 
nates is  the  same,  wherever  on  this 
line  the  point  P  be  taken.  This  ratio 
is  equal  to  the  tangent  of  the  angle  a  ^ 
made  by  the  line   with  the  axis  Ox,  Fig.  23 

i.e,  to  what  we.  shall  call  the  slope  of  the  line.     Let  us  put 

tan  a  =  w ; 
then  we  have,  for  any  point  P  on  this  line  :  y/x  =  m,  i.e. : 
(1)  y  =  mx. 

Moreover,  for  any  point  Q,  not  on  this  line,  the  ratio  y/x 
must  evidently  be  different  from  tan  a,  i.e.  from  m.  The  equa- 
tion y  =  mx  is  therefore  said  to  represent  the  line  through  O 
whose  slope  is  m;  and  y  =  7nx  is  called  the  equation  of  this  line. 
We  mean  by  this  statement  that  the  relation  y  =  mx  is  satis- 
fied by  the  coordinates  of  every  point  on  the  line  OP,  and  only 
by  the  coordinates  of  the  points  on  this  line.  Notice  in  partic- 
ular that  the  coordinates  of  the  origin  0,  i.e.  x  =  0,  y  =  0, 
satisfy  the  equation  y  =  mx. 

24.  Proportional  Quantities.  Any  two  values  of  x  are 
proportional  to  the  corresponding  values  of  y  it  y  =  mx.  For, 
if  (xi ,  ?/i)  and  (iCg ,  2/2)  ^^^  two  pairs  of  values  of  x  and  y  that 
satisfy  (1),  we  have 

yi==mxi,   y2  =  mx2; 


*  For  the  sake  of  brevity,  a  straight  line  will  here  in  general  be  spoken  oi 
simply  as  a  line ;  a  line  that  is  not  straight  will  be  called  a  curve. 


II,  §  24]  THE  STRAIGHT  LINE  25 

hence,  dividing, 

2/1/2/2  =  Va?2. 

The  constant  quantity  m  is  called  the  factor  of  proportionality. 

Many  instances  occur  in  mathematics  and  in  the  applied 
sciences  of  two  quantities  related  to  each  other  in  this  man- 
ner. It  is  often  said  that  one  quantity  y  varies  as  the  other 
quantity  x. 

Thus  Hooke's  Law  states  that  the  elongation  E  of  a,  stretched 
wire  or  spring  varies  as  the  tension  t ;  that  is,  E  =  Jet,  where  k 
is  a  constant. 

Again,  the  circumference  c  of  a  circle  varies  as  the  radius  r; 

EXERCISES 

1.  Draw  each  of  the  lines  : 

(a)y  =  2x.         (c)    yz=-j\x.         (e)    5x+3?/=0.  {g)y  =  -x. 

(b)  y=~Sx.       {d)5y  =  Sx.  {f)y  =  x.  (h)x-y  =  Q. 

2.  Show  that  the  equation  ax  -^-hy  =Q  can  be  reduced  to  the  form 
y  =  7nx,  if  &  :51b  0,  and  therefore  represents  a  line  through  the  origin. 

3.  Find  the  slope  of  the  lines : 

(a)  x  +  y=0.  (c)  Sx_-^y  =  0. 

(b)  x-y  =  0.  (d)   \/2x  +  y  =  0. 

4.  Draw  a  line  to  represent  Hooke's  Law  E  =  kt,  ii  k  =  10  (see  Ex.  7, 
p.  15).  Let  t  be  represented  as  horizontal  lengths  (as  is  x  in  §  23)  and 
let  E  be  represented  by  vertical  lengths  (as  is  ?/  in  §  23). 

6.  Draw  a  line  to  represent  the  relation  c  =  2  7rr,  where  c  means  the 
circumference  and  r  the  radius  of  a  circle. 

6.  The  number  of  yards  y  in  a  given  length  varies  as  the  number  of 
feet  /  in  the  same  length  ;  in  particular,  f=Sy.  Draw  a  figure  to 
represent  this  relation. 

7.  If  1  in.  =  2.54  cm.,  show  that  c  =  2.54  i,  where  c  is  the  number  of 
centimeters  and  i  is  the  number  of  inches  in  the  same  length.  Draw  a 
figure. 


26 


PLANE  ANALYTIC  GEOMETRY 


[11,  §  25 


25.  Slope  Form.  Finally,  consider  a  line  that  does  not  pass 
through  the  origin  and  is  not  parallel  to  either  of  the  axes  of 
coordinates  (Fig.  24) ;  let  it  intersect  the  axes  Ox,  Oy  at  A, 
B,  respectively,  and  let  P(x,  y)  be  any  other  point  on  it.  The 
figure  shows  that  the  slope  m  of  y 

the  line,  i.e.  the  tangent  of  the 
angle  a  at  which  the  line  is  in-  b 

clined  to  the  axis  Ox,  is  ^^^^ 

RP 


m  =  tan  a  = 


or,  since  i2P=  QP 


BR' 
-QR=QP-OB 

y  —  b 


Fig.  24 


-bSiXidBR  =  OQ=:x: 


that  is, 

(2)  y  =  mx  4-  b, 

where  b  =  OB  is  called  the  intercept  made  by  the  line  on  the 

axis  Oy,  or  briefly  the  y-intercept. 

The  slope  angle  a  at  which  the  line  is  inclined  to  the  axis  Ox 
is  always  understood  as  the  smallest  angle  through  which  the 
positive  half  of  the  axis  Ox  must  be  turned  counterclockwise 
about  the  origin  to  become  parallel  to  the  line. 

26.  Equation  of  a  Line.  On  the  line  AB  oi  Fig.  24  take 
any  other  point  P' ;  let  its  coordinates  be  x',  y',  and  show  that 
y'  =  mx'  +  b. 

Take  the  point  P'  {x',  y')  outside  the  line  AB  and  show  that 
the  equation  y  =  mx  +  6  is  not  satisfied  by  the  coordinates  x', 
y'  of  such  a  point. 

For  these  reasons  the  equation  ys=mx-\-b  is  said  to  represent 
the  line  ivhose  y-intercept  is  b  and  ivhose  slope  is  m ;  it  is  also 
called  the  equation  of  this  line.  The  ^/-intercept  OB  =  b  and 
the  slope  m  =  tan  a  together  fully  determine  the  line. 


II,  §  26]  THE  STRAIGHT  LINE  27 

Every  line  of  the  plane  can  be  represented  by  an  equation  of  the 

form 

y  =  mx  +  b, 

excepting  the  lines  parallel  to  the  axis  Oy.  When  the  line  be- 
comes parallel  to  the  axis  Oy,  both  its  slope  m  and  its  ly-inter- 
cept  b  become  infinite.  We  have  seen  in  §  22  that  the  equa- 
tion of  a  line  parallel  to  the  axis  Oy  is  of  the  f  brm  x  —  a. 

Eeduce  the  equation  ^x—2y=5  to  the  form  y  =  mx-\-b  and 
sketch  the  line. 

EXERCISES 

1.  Sketch  the  lines  whose  y-intercept  is  &  =  2  and  whose  slopes  are 
m  =  I,  3,  0,  —  I ;  write  down  their  equations. 

2.  Sketch  the  lines  whose  slope  is  w  =  4/3  and  whose  ^/-intercepts  are 
0,  1,  2,  5,  —  1,  —  2,  —  6,  —  12.2,  and  write  down  their  equations. 

3.  Sketch  the  lines  whose  equations  are : 

(a)  y=2x+S.        (c)  y=x-l.     (e)   x-y=l.  (g)  1x-y  +  l2=0. 

(6)  y=_ix+l.     (d)x-\-y  =  l.     (/)x-2y  +  2=0.     (/i)  4x  +  3?/  +  5=0. 

4.  Do  the  points  (1,  5),  (-2,  -1),  (3,  7)  lie  on  the  line  y  =  2x-\-Z  ? 

5.  A  cistern  that  already  contained  300  gallons  of  water  is  filled  at  the 
rate  of  1 00  gallons  per  hour.  Show  that  the  amount  A  of  water  in  the 
cistern  n  hours  after  filling  begins  is  J.  =  100  w+300.  Draw  a  figure  to 
represent  this  relation,  plotting  the  values  of  A  vertically,  with  1  vertical 
space  =  100  gallons. 

6.  In  experiments  with  a  pulley  block,  the  pull  p  in  lbs.,  required  to 
lift  a  load  I  in  lbs.,  was  found  to  be  expressed  by  the  equation  p  =  .  15  Z  + 2. 
Draw  this  line.  How  much  pull  is  required  to  operate  the  pulley  with  no 
load  (i.e.  when  1  =  0)? 

7.  The  readings  of  a  gas  meter  being  tested,  T,  were  found  in  compari- 
son with  those  of  a  standard  gas  meter  S,  and  the  two  readings  satisfied 
the  equation  r  =  300  +  1.2  S.  Draw  a  figure.  What  was  the  reading 
T  when  the  reading  S  was  zero  ?  What  is  the  meaning  of  the  slope  of 
the  line  in  the  figure  ? 


28  PLANE  ANALYTIC  GEOMETRY  [II,  §  27 

27.   Parallel  and  Perpendicular  Lines.     Two  lines 
y  =  m-^x  +  &i ,    2/  =  '^^2^*  +  ^2 
are  obviously  parallel  if  they  have  the  same  slope,  i.e.  if 

(3)  mi  =  m^. 

Two  lines  2/  =  mjcc  4-  ft^ ,  ?/  =  mga;  +  h^  are  perpendicular  if  the 
slope  of  one  is  equal  to  minus  the  reciprocal  of  the  slope  of 
the  other,  i.e.  if 

(4)  mima  =  —  1. 

For  if  m2  =  tan  aj ,  mg  =  tan  Wg ,  the  condition  that  mim^  =  —  1 
gives  tan  aa  =  —  1/tan  ctj  =  —  cot  a^ ,  whence  ots  =  «i  +  i  t. 

(^C  EXERCISES 

1.  Write  down  the  equation  of  any  line  :  (a)  parallel  to  y  =  3  a:  —  2, 
(6)  perpendicular  to  y  =  3  x  —  2. 

2.  Show  that  the  parallel  to  y  =  Sx  —  2  through  the  origin  isy  =  S x. 

3.  Show  that  the  perpendicular  to  y  =zSx  —  2  through  the  origin  is 

y=-^x. 

4.  For  what  value  of  b  does  the  line  y  =  Sx  +  b  pass  through  the 
point  (4,  1)  ?    Find  the  parallel  to  ?/  =  3  x  —  2  through  the  point  (4,  1). 

6.    Find  the  parallel  to  y  =  5x  +  1  through  the  point  (2,  3). 

6.  Find  the  perpendicular  to  y  =  2x  —  1  through  the  point  (1,  4). 

7.  What  is  the  geometrical  meaning  of  61  =  62  in  the  equations 

y  —  m-iX  +  ?>i ,   y  =  m^x  +  &2  ? 

8.  Two  water  meters  are  attached  to  the  same  water  pipe  and  the  water 
is  allowed  to  flow  steadily  through  the  pipe.  The  readings  B\  and  ^2  of  the 
two  meters  are  found  to  be  connected  with  the  time  t  by  means  of  the 
equations  Bi  =  2.6t,   i?2  =  2.5«  +  150, 

where  i?i  and  B2  are  measured  in  cubic  feet  and  t  is  measured  in  seconds. 
Show  that  the  lines  that  represent  these  equations  are  parallel.  What 
is  the  meaning  of  this  fact  ? 

9.  The  equations  connecting  the  pull  p  required  to  lift  a  load  lo  is 
found  for  two  pulley  blocks  to  be 

pi  =  .05  w;  -t-  2,  p2  =  .05  w  +  1.6 
Show  that  the  lines  representing  these  equations  are  parallel.    Explain. 


II,  §29]  THE  STRAIGHT  LINE  29 

10.  The  equations  connecting  the  pull  p  required  to  lift  a  load  w  is 
found  for  two  pulley  blocks  to  be 

Pi  =  .1510  +  1.5,  p<i  —  .05  w  +  1.5. 

Show  that  the  lines  representing  these  equations  are  not  parallel,  but 
that  the  values  of  pi  and  p-i  are  equal  when  lo  =  0.     Explain. 

28.  Linear  Function.  The  equation  y  =  mx+b,  when  m 
and  b  are  given,  assigns  to  every  value  of  x  one  and  only  one 
definite  value  of  y.  This  is  often  expressed  by  saying  that 
mx  +  6  is  a  function  of  x ;  and  as  the  expression  mx  +  6  is  of 
the  first  degree  in  x,  it  is  called  Siftinctiori  of  the  first  degree  or, 
owing  to  its  geometrical  meaning,  a  linear  function  of  x. 

Examples  of  functions  of  x  that  are  not  linear  are  3  ic^  —  5, 
ax^  -\-hx-\-c,  x{x  —  l),  1/x,  sin  a;,  10"^,  etc.  The  equations 
y  =  3a^  —  5,  y  =  ax^ -{- bx -\- Cj  etc.,  represent,  as  we  shall  see 
later,  not  straight  lines  but  curves. 

The  linear  function  y  =  mx  +  b,  being  the  most  simple  kind 
of  function,  occurs  very  often  in  the  applications.  Notice  that 
the  constant  b  is  the  value  of  the  function  for  x  =  0.  The  con- 
stant m  is  the  rate  of  change  of  y  with  respect  to  x. 

29.  Illustrations.  Example  1.  A  man,  on  a  certain  date, 
has  $10  in  bank;  he  deposits  $3  at  the  end  of  every  week; 
how  much  has  he  in  bank  x  weeks  after  date  ? 

Denoting  by  y  the  number  of  dollars  in  bank,  we  have 

y  =  3x-\-10. 

His  deposit  at  any  time  a;  is  a  linear  function  of  x.  Notice 
that  the  coefficient  of  x  gives  the  rate  of  increase  of  this  de- 
posit ;  in  the  graph  this  is  the  slope  of  the  line. 

Example  2.  Water  freezes  at  0°  C.  and  32°  F. ;  it  boils  at 
100'*  C.  and  at  212°  F. ;  assuming  that  mercury  expands  uni- 
formly, i.e.  proportionally  to  the  temperature,  and  denoting 


30  PLANE  ANALYTIC   GEOMETRY  [II,  §  29 

by  X  any  temperature  in  Centigrade  degrees,  by  y  the  same 
temperature  in  Fahrenheit  degrees,  we  have 

y-S2     212-32     9     .  o      .  oo 

If  the  line  represented  by  this  equation  be  drawn  accurately, 
on  a  sufficiently  large  scale,  it  could  be  used  to  convert  centi- 
grade temperature  into  Fahrenheit  temperature,  and  vice  versa. 

Example  3.  A  rubber  band,  1  ft.  long,  is  found  to  stretch 
1  in.  by  a  suspended  mass  of  1  lb.  Let  the  suspended  mass 
be  increased  by  1  oz.,  2  oz.,  etc.,  and  let  the  corresponding 
lengths  of  the  band  be  measured.  Plotting  the  masses  as  ab- 
scissas and  the  lengths  of  the  band  as  ordinates,  it  will  be 
found  that  the  points  (x,  y)  lie  very  nearly  on  a  straight  line 
whose  equation  is  y  =  ^^x  -\-l.  The  experimental  fact  that 
the  points  lie  on  a  straight  line,  i.e.  that  the  function  is  linear, 
means  that  the  extension,  y  —  1,  is  proportional  to  the  tension j 
i.e.  to  the  weight  of  the  suspended  mass  x  (Hooke's  Law). 

Notice  that  only  the  part  of  the  line  in  the  first  quadrant, 
and  indeed  only  a  portion  of  this,  has  a  physical  meaning. 
Can  this  range  be  extended  by  using  a  spiral  steel  spring  ? 

Example  4.  When  a  point  P  moves  along  a  line  so  as  to 
describe  always  equal  spaces  in  equal  times,  its  motion  is  called 
uniform.  The  spaces  x^assed  over  are  then  proportional  to  the 
times  in  which  they  are  described,  and  the  coefficient  of  pro- 
portionality, i.e.  the  ratio  of  the  distance  to  the  time,  is  called 
the  velocity  v  of  the  uniform  motion.  If  at  the  time  t  =  0  the 
moving  point  is  at  the  distance  Sq,  and  at  the  time  t  at  the  dis- 
tance s,  from  the  origin,  then 

S  =  SQ-\-Vt. 

Thus,  in  uniform  motion,  the  distance  s  is  a  linear  function  of 
the  time  t,  and  the  coefficient  of  t  is  the  speed :  v  =  (s  —  SQ)/t. 


II,  §  29]  THE  STRAIGHT  LINE  31 

Example  5.  When  a  body  falls  from  rest  (in  a  vacuum)  its 
velocity  v  is  proportional  to  the  time  t  of  falling :  v=  gt,  where 
g  is  about  32  if  the  velocity  is  expressed  in  ft./sec,  or  980 
if  the  velocity  is  expressed  in  cm./sec. 

If,  at  the  time  t  =  0,  the  body  is  thrown  downward  with  an 
initial  velocity  Vq,  its  velocity  at  any  subsequent  time  t  is 

v  =  Vq  +  gt. 
Thus  the  velocity  is  a  linear  function  of  t,  and  the  coefficient  g 
of  t  denotes  the  rate  at  which  the  velocity  changes  with  the 
time,  i.e.  the  acceleration  of  the  falling  body. 

EXERCISES 

1.  Draw  the  line  represented  by  the  equation  y  =  f  x  +  32  of  Ex- 
ample 2,  §  29.  What  is  its  slope  ?  What  is  the  y-intercept  ?  What  is 
the  meaning  of  each  of  these  quantities  if  y  and  x  represent  the  tempera- 
tures in  Fahrenheit  and  in  Centigrade  measure,  respectively  ? 

2.  Kepresent  the  equation  ?/  =  j^^  a:  +  1  of  Example  3,  §  29,  by  a  figure. 
What  is  the  meaning  of  the  ?/-intercept  ? 

3.  Draw  the  line  s  =  sq  -\-  vt  of  Example  4,  §  29,  for  the  values  Sq  =  10, 
^  =  3.  What  is  the  meaning  ofv?  Show  that  the  speed  v  may  be  thought 
of  as  the  rate  of  increase  of  s  per  second. 

4.  If,  in  the  preceding  exercise,  v  be  given  a  value  greater  than  3, 
how  does  the  new  line  compare  with  the  one  just  drawn  ? 

6.  If,  in  Ex.  3,  v  is  given  the  value  3,  and  so  several  different  values, 
show  that  the  lines  represented  by  the  equation  are  parallel.     Explain. 

6.  In  experiments  on  the  temperatures  at  various  depths  in  a  mine, 
the  temperature  (Centigrade)  T  was  found  to  be  connected  with  the 
depth  d  by  the  equation  r=  60  +  .01  d,  where  d  is  measured  in  feet. 
Draw  a  figure  to  represent  this  equation.  Show  that  the  rate  of  increase 
of  the  temperature  was  1°  per  hundred  feet. 

7.  In  experiments  on  a  pulley  block,  the  pull  p  (in  lb.)  required  to 
lift  a  weight  w  (in  lb.)  was  found  to  he  p  =  .03  w  ■{■  0.5.  Show  that  the 
rate  of  increase  of  p  is  3  lb.  per  hundred  weight  increase  in  w. 


32  PLANE  ANALYTIC  GEOMETRY  [II,  §  30 

30.    General  Linear  Equation.     The  equation 

in  which  A,  B,  C  are  any  real  numbers,  is  called  the  general 
equation  of  the  first  degree  in  x  and  y.  The  coefficients  Ay  B,  C 
are  called  the  constants  of  the  equation ;  x,  y  are  called  the 
variables.  It  is  assumed  that  A  and  B  are  not  both  zero. 
The  terms  Ax  and  By  are  of  the  first  degree ;  the  term  C  is 
said  to  be  of  degree  zero  because  it  might  be  written  in  the 
form  Cx^ ;  this  term  C  is  also  called  the  constant  term. 
Every  equation  of  the  first  degree, 

(5)  Ax+By-\-C  =  0, 

in  which  A  and  B  are  not  both  zero,  represents  a  straight  line; 
and  conversely,  every  straight  line  can  be  represented  by  such  an 
equation.  For  this  reason,  every  equation  of  the  first  degree 
is  called  a  linear  equation. 

The  first  part  of  this  fundamental  proposition  follows  from 
the  fact  that,  when  B  is  not  equal  to  zero,  the  equation  can  be 
reduced  to  the  form  y  =  mx-^  bhy  dividing  both  sides  by  B ; 
and  we  know  that  y  =  mx  -\-  b  represents  a  line  (§  25).  When 
B  is  equal  to  zero,  the  equation  reduces  to  the  form  x  =  a, 
which  also  represents  a  line  (§  22). 

The  second  part  of  the  theorem  follows  from  the  fact  that 
the  equations  which  we  have  found  in  the  preceding  articles 
'for  any  line  are  all  particular  cases  of  the  equation 

Ax-\-By  -\-  C  =  0. 
This  equation  still  expresses  the  same  relation  between  x 
and  y  when  multiplied  by  any  constant  factor,  not  zero.  Thus, 
any  one  of  the  constants  A,  B,  C,  if  not  zero,  can  be  reduced 
to  1  by  dividing  both  sides  of  the  equation  by  this  constant. 
The  equation  is  therefore  said  to  contain  only  tivo  (not  three) 
essential  constants. 


II,  §  32] 


THE  STRAIGHT  LINE 


33 


31.   Conditions  for  Parallelism  and  for  Perpendicularity. 

It  is  easy  to  recognize  whether  two  lines  whose  equations  are 
Ax  +  By-i-C=0  and  A'x -\-  B'y  +  C"  =  0  are  parallel  or  per- 
pendicular. The  lines  are  parallel  if  they  have  the  same  slope, 
and  they  are  perpendicular  (§  27)  if  the  product  of  their  slopes 
is  equal  to  —1.  The  slopes  of  our  lines  are  —  A/B  and 
—  A'/B' ;  hence  these  lines  are  parallel  if  —  A/B  =  —  A'/B'y 

*-e-if  A:B  =  A':B': 


and  they  are  perpendicular  if 

^.^;  =  -l,  Le.ii 

B    B'  ' 


AA'  +  BB'  =  0. 


32.  Intercept  Form.  If  the  constant  term  (7  in  a  linear 
equation  is  zero,  the  equation  represents  a  line  through  the 
origin.  For,  the  coordinates  (0,  0)  of  the  origin  satisfy  the 
equation  Ax  +  By  =  0. 

If  the  constant  terra  C  is  not  equal  to  zero,  the  equation 
Ax  +  By  -\-  C  =  0  can  be  divided  by  (7 ;  it  then  reduces  to  the 


form 


^x  +  |,  +  l=0. 


If  A  and  B  are  both  different  from  zero,  this  can  be  written : 


+ 


y 


-  C/A  '   -  C/B       ' 
or  putting  —  C/A  =  a,  —  C/B 

(6) 


6: 


a      o 


Fig.  25 


The  conditions  A^(),  B^^O  mean 
evidently  that  the  line  is  not  parallel  to  either  of  the  axes. 
Therefore  the  equation  of  any  line,  not  passing  through  the 
origin,  and  not  parallel  to  either  axis,  can  be  written  in  the 


34  PLANE  ANALYTIC   GEOMETRY  [II,  §  32 

form  (6).  With  2/ =  0  this  equation  gives  a;  =  a;  with  x  =  0 
it  gives  y  =  b.     Thus 

^         B'  A 

are  the  intercepts  (Fig.  25)  made  by  the  line  on  the  axes  Oxy 
Oyj  respectively  (see  §  25). 

EXERCISES 

1.  Write  down  the  equations  of  the  line  whose  intercepts  on  the 
axes  Ox^  Oy  are  5  and  —  3,  respectively  ;  the  line  whose  intercepts  are 

—  I  and  7  ;  the  line  whose  intercepts  are  —  1  and  —  |.  Sketch  each  of 
the  lines  and  reduce  each  of  the  equations  to  the  form  Ax-\-By-{-C=0,  so 
that  A,  B,  C  are  integers. 

2.  Find  the  intercepts  of  the  lines :    Sx  —  2y  =  1,  x  +  ly-{-l  =  0, 

—  Sx  +  ^y  —  5  =  0.  Try  to  read  off  the  values  of  the  intercepts  directly 
from  these  equations  as  they  stand. 

■  3.   In  Ex.  2,  find  the  slopes  of  the  lines. 

4.  Prove  (6),  §  32  by  equality  of  areas,  after  clearing  of  fractions. 

5.  What  is  the  equation  of  the  axis  Oy  ?  of  the  axis  Ox  ? 

6.  What  is  the  value  of  B  such  that  the  line  represented  by  the  equa- 
tion ix-{-  By  —  li  =  0  passes  through  the  point  (—  5,  17)  ? 

7.  What  is  the  value  of  A  such  that  the  line  Ax  -\-  7  y  =  10  has  its 
OS-intercept  equal  to  —  8  ? 

8.  Reduce  each  of  the  following  equations  to  the  intercept  form  (6), 
and  draw  the  lines  : 

(a)  Sx-6y-16  =  0.  (b)  x -{- ^y +  1=  0. 

(c)    i^-3y-6^^  (d)  5.x  =  3x  +  ?/-10. 

x  +  y 

9.  Reduce  the  equations  of  Ex.  8  to  the  slope  form  (2),  §  25. 

10.  Find  the  equation  of  the  hue  of  slope  0  passing  through  the  point 
(6,-5). 


n,  §32]  THE  STRAIGHT  LINE  35 

XI.  What  relation  exists  between  the  coefficients  of  the  equation 
Ax-i-  By  +  C  =  0,  ii  the  line  is  parallel  to  the  line  ix  —  6y  =  8?  parallel 
to  the  axis  Oy  ? 

12.  Show  that  the  points  (-  1,  -7),  Q,  -3),  (2,2),  (-2,  -10) 
lie  on  the  same  line. 

13.  Find  the  area  of  the  triangle  formed  by  the  lines  x+y=0,  x—y=0, 
X—  a  =  0. 

14.  Show  that  the  line  4(ic  —  a)  +  5(y  —  &)  =  0  is  perpendicular  to  the 
line  5  ic— 4  y—10=0  and  passes  through  the  point  (a,  b). 

15.  A  line  has  equal  positive  intercepts  and  passes  through  (—5,  14). 
What  is  its  equation  ?  its  slope  ? 

16.  If  a  line  through  the  point  (6,  7)  has  the  slope  4,  what  is  its 
y-intercept  ?  its  a;-intercept  ? 

17.  The  Reaumur  thermometer  is  graduated  so  that  water  freezes  at 
0°  and  boils  at  80".  Draw  the  line  that  represents  the  reading  B  of  the 
Reaumur  thermometer  as  a  function  of  the  corresponding  reading  G  of 
the  Centigrade  thermometer. 

18.  What  function  of  the  altitude  is  the  area  of  a  triangle  of  given 
base  ? 

19.  A  printer  asks  75  f  to  set  the  type  for  a  program  and  2  ^  per  copy 
for  printing.  The  total  cost  is  what  function  of  the  number  of  copies 
printed  ?    Draw  the  line  representing  the  function. 

Another  printer  asks  3  ^  per  copy,  with  no  charges  for  setting  the  type. 
For  how  many  copies  would  both  charge  the  same  ? 

20.  The  sum  of  two  complementary  angles  a  and  j3  is  ^  tt  ;  draw  the 
line  representing  /3  as  a  function  of  a.     When  a  =  |  tt,  what  is  /3  ? 

21.  Express  the  value  of  a  note  of  §  1000  at  the  end  of  the  first  year  as 
a  function  of  the  rate  of  interest.  At  6%  simple  interest  its  value  is  what 
function  of  the  time  in  years  ? 

22.  Two  weights  are  attached  to  the  opposite  ends  of  a  rope  that  runs 
through  a  double  pulley  block  of  which  one  block  is  fastened  at  a  height 
above  ground.  If  x  and  y  denote  the  distances  of  the  two  weights  above  the 
ground,  determine  a  linear  relation  between  them  if  a;  =  40  when  y  =  0 
and  y  =  10  when  x  =  0. 


36  PLANE  ANALYTIC  GEOMETRY  [II,  §  33 

33.  Line  through  One  Point.  To  find  the  line  of  given 
slope  7ni  through  a  given  point  Pi(i«i,  2/1)?  observe  that  the 
equation  must  be  of  the  form  (2),  viz. 

y  =  TiiiX  +  b, 

since  this  line  has  the  slope  m^.  If  this  line  is  to  pass  through 
the  given  point,  the  coordinates  x^,  y^  must  satisfy  this  equa- 
tion, i.e.  we  must  have 

2/1  =  ra^^i  +  &• 
This  equation  determines  h,  and  the  value  of  h  so  found  might 
be  substituted  in  the  preceding  equation.     But  we  can  eliminate 
h  more  readily  between  the  two  equations  by  subtracting  the 
latter  from  the  former.     This  gives 

y-yi  =  m,{x-x,) 

as  the  equation  of  the  line  of  slope  Wj  through  Pi{xi,  y^. 

The  problem  of  finding  a  line  through  a  given  point  parallel, 
or  perpendicular,  to  a  given  line  is  merely  a  particular  case  of 
the  problem  just  solved,  since  the  slope  of  the  required  line  can 
be  found  from  the  equation  of  the  given  line  (§  27).  If  the 
slope  of  the  given  line  is  m^  =  tan  a-^,  the  slope  of  any  parallel 
line  is  also  mj,  and  the  slope  of  any  line  perpendicular  to  it  is 

mg  =  tan  (ai-\-lir)  =  —  cot  a^  =  —  — . 

mi 

34.  Line  through  Two  Points.  To  find  the  line  through  two 
given  points,  Pi{xi,  2/0?  -^2(^2?  2/2)?  observe  (Fig.  26)  that  the 
slope  of  the  required  line  is  evi-  ^ 
dently 

if,  as  in  §  9,  we  denote  by  A  a;,  A?/ 
the  projections  of  P1P2  011  Ox,  Oy\ 


II,  §  34]  THE  STRAIGHT  LINE  37 


and  as  the  line  is  to  pass  through  (xi,  y^),  we  find  its  equation 

by  §  33  as  V    V  ^  ^.  V ,  ^  j^^ ,;  ^p^ 


2/2-2/1,   . 

—  W,  = (x  —  cc, ), 


a;    2/ 

1 

^'1   2/1 

1 

^2  2/2 

1 

y-yi  =  -~(^-^i)' 

The  equation  of  the  line  through  two  given  points  (aj^,  yi), 
fe  2/2)  can  also  be  written  in  the  determinant  form 

which  (§  14)  means  that  the  point  (x,  y)  is  such  as  to  form 
with  the  given  points  a  triangle  of  zero  area.  By  expanding 
the  determinant  it  can  be  shown  that  this  equation  agrees  with 
the  preceding  equation.  A  more  direct  proof  will  be  given 
later  (§  49). 

EXERCISES 

1.  Find  the  equation  of  the  line  through  the  point  (—7,  2)  parallel 
to  the  line  y  =  Sx. 

2.  Show  that  the  points  (4,  —3),  (—5,  2),  (5,  20)  are  the  vertices  of 
a  right  triangle. 

3.  Find  the  equation  of  the  line  through  the  point  (—  6,  —  3)  which 
makes  an  angle  of  30°  with  the  axis  Ox  ;  30°  with  the  axis  Oy. 

4.  Does  the  line  of  slope  |  through  the  point  (4,  3)  pass  through  the 
.point  (—5,  -4)  ? 

5.  Find  the  equation  of  the  line  through  the  point  (—2,  1)  parallel  to 
the  line  through  the  points  (4,  2)  and  (-  3,  —  2). 

6.  Find  the  equations  of  the  lines  through  the  origin  which  trisect 
that  portion  of  the  line  5  x  -  6  y  =  60  which  lies  in  the  fourth  quadrant. 

7.  What  are  the  intercepts  of  the  line  through  the  points  (2,  —3), 
(-5,  4)  ? 


38  PLANE  ANALYTIC  GEOMETRY  [II,  §  34 

8.  Show  that  the  equation  of  the  line  through  the  point  (a,  6)  per- 
pendicular to  the  line  Ax-\-  By  -{-  C  =  0  is  (x  —  a)/ A  =  (y  —  h)/B. 

9,  Find  the  equations  of  the  diagonals  of  the  rectangle  formed  by  the 
lines  ic  +  a  =  0,  a;  —  &=0,  ?/  +  c  =  0,  y  —  cZ  =  0. 

10.  Find  the  equation  of  the  perpendicular  bisector  of  the  line  joining 
the  points  (4,  —5)  and  (—  3,  2).  Show  that  any  point  on  it  is  equally 
distant  from  each  of  the  two  given  points. 

11.  Find  the  equation  of  the  line  perpendicular  to  the  line  4x— 3?/+6=0 
that  passes  through  the  midpoint  of  (—4,  7)  and  (2,  2). 

12.  What  are  the  coordinates  of  a  point  equidistant  from  the  points 
(2,  —3)  and  (  —  5,  0)  and  such  that  the  line  joining  the  point  to  the  origin 
has  a  slope  1  ? 

13.  If  the  axes  are  oblique  with  angle  t»7,  show  that  the  slope  of  the 
line  joining  the  points  P\(x\,  y\)  and  P^ixo,  y^)  is 

(y2  —  yi)  sin  a> 

(a;2-a;i)  +  {y2-yi)cosw 

^  14.  If  the  axes  are  oblique  with  angle  w,  show  that  the  equation  of  the 
line  through  the  point  Pi{xi,  yi)  which  makes  with  the  axis  Ox  the 
angle  (p,  is 

sm  (w  —  0) 
Is  the  coefficient  of  (x  —  Xi)  the  slope  of  this  line  ? 

15.  In  an  experiment  with  a  pulley-block  it  is  assumed  that  the  rela- 
tion between  the  load  I  and  the  pull  p  required  to  lift  it  is  linear.  Find 
the  relation  Up  =  8  when  I  =  100,  and p  =  12  when  I  =  200. 

16.  In  an  experiment  in  stretching  a  brass  wire  it  is  assumed  that  the 
elongation  E  is  connected  with  the  tension  t  by  means  of  a  linear  relation. 
Find  this  relation  if  t  =  IS  lb.  when  E  =  .1  in.,  and  i  =  58  lb.  when 
^  =  .3  in. 

17.  A  cistern  is  being  filled  by  water  flowing  into  it  at  the  rate  of  30 
gallons  per  second.  Assuming  that  the  amount  A  of  water  in  the  cistern 
is  connected  with  the  time  «  by  a  hnear  relation,  find  this  relation  if 
A  =  1000  when  (  =  10.     Hence  find  A  when  t  =  0. 


CHAPTER   III 

SIMULTANEOUS    LINEAR    EQUATIONS 
DETERMINANTS 

PART  I.     EQUATIONS  IN  TWO   UNKNOWNS 
DETERMINANTS   OF   SECOND   ORDER 

35.  Intersection  of  Two  Lines.  The  point  of  intersection 
of  any  two  lines  is  found  by  solving  the  equations  of  the  lines  as 
simultaneous  equations.  For  the  coordinates  of  the  point  of 
intersection  must  satisfy  each  of  the  two  equations,  since  this 
point  lies  on  each  of  the  two  lines ;  and  it  is  the  only  point 
having  this  property.  Find  the  points  of  intersection  of  the 
following  pairs  of  lines : 

^^^    l3a;  +  52/-34  =  0.  ^^    \lx  +  2y=^0.    . 

^^^  l5a;-22/  +  ll  =  0. 
The  solution  of  simultaneous  linear  equations  is  much 
facilitated  by  the  use  of  determinants.  As,  moreover,  deter- 
minants are  used  to  advantage  in  many  other  problems  (see, 
e.g.,  §§  12,  14)  it  is  desirable  to  study  determinants  systemati- 
cally before  proceeding  with  the  study  of  the  straight  line. 

36.  Solution  of  Two  Linear  Equations.  To  solve  two 
linear  equations  (§  30), 

[  ttga;  +  &^  =  ^2  J 
we  may  eliminate  y  to  find  x,  and  eliminate  x  to  find  y.     The 
elimination  of  y  is  done  systematically  by  multiplying  the  first 

39 


40 


PLANE  ANALYTIC  GEOMETRY         [III,  §  36 


equation  by  63?  the  second  by  61,  and  then  subtracting  the 
second  from  the  first ;  this  gives 

Likewise,  to  eliminate  x,  multiply  the  first  equation  by  a2,  the 
second  by  aj ,  and  subtract  the  first  from  the  second  : 

If  aib2  —  a2&i  =^  ^}  we  can  divide  by  this  quantity  and  thus 
find 

/f)\                                          ^         "'1"2  —  ^2^1        „.         ^V^2  —  ^2% 
(Z)  X  =  — ; —  ,     y  = ; ,     • 

^  a^2  —  «20i  otiOa  —  «20i 

Observe  that  the  values  of  x  and  y  are  quotients  with  the 
same  denominator,  and  that  the  numerator  of  x  is  obtained 
from  this  denominator  by  simply  replacing  a  by  A;,  while  the 
numerator  of  y  is  obtained  from  the  same  denominator  by 
replacing  h  by  k. 

This  peculiar  form  of  the  numerators  and  denominators  of 
X  and  y  is  brought  out  more  clearly  if  we  agree  to  write  the 
common  denominator  ajftg  —  ^2^1  in  the  form  of  a  determinant : 

an       60 


(3) 

as  in  §  12. 


Thus 


2 

7 
-1 
4 


=2x5-7x3 


11; 


=  -lx2-4x7=-30. 


With  this  notation,  the  values  (2)  of  x  and  y  are 


(4) 


a;  = 


^1     &i 

"'2       ^2 


% 

fc, 

aj 

fc, 

Oi 

61 

aj 

6. 

I" 


Ill,  §  37]    SIMULTANEOUS  LINEAR  EQUATIONS 


41 


37.  General  Rule.  If  a,  h,  c,  d  are  any  four  numbers,  the 
expression  a    i,  i 

c   ay 

which  stands  for  ad  — he,  is  called  a  determinant,  more  pre- 
cisely, a  determinant  of  the  second  order  because  two  numbers 
occur  in  each  (horizontal)  row,  as  well  as  in  each  (vertical) 
column.     (See  §  12.) 

The  determinant  (3)  is  called  the  determinant  of  the  equa- 
tions (1),  §  36. 

We  can  then  state  the  following  rule  for  solving  the  two 
linear  equations  (1) :  If  the  determinant  of  the  equations  is  not 
equal  to  zero,  x  as  well  as  y  is  the  quotient  of  two  determinants ; 
the  denominator  is  the  sayiie,  viz.  the  determinant  of  the  equa- 
tio7is  (1)  ;  the  numerator  of  x  is  obtained  from  this  denominator 
by  replacing  the  coefficients  of  x  by  the  constant  terms,  the  numer- 
ator of  y  is  found  from  the  same  denominator  by  replacing  the 
coefficients  ofy  by  the  constant  terms.* 


EXERCISES 

1.   Find  the  values  of  the  following  determinants  : 


(«) 

10    2 

3    7 

(c?) 

0    0 

12     5 

(0 


2-1 
1         2 
h     -12 
f       -§ 

2.   Solve  the  following  equations ;  in  writing  down  the  solution,  begin 
with  the  denominators : 


(a-)    P^-2y  =  l, 

|2x  +  3y  +  4  =  0, 
^  ^    \Bx-5y-16  =  0. 


(ft) 


i2x  +  7y  =  3, 
\5x-y  =  -ll. 
l6x-3y-2  =  0, 
[y  =:4x-l. 


*  One  great  advantage  of  this  rule  is  that  the  same  rule  applies  to  the  solu- 
tion of  any  (finite)  number  of  linear  equations  with  the  same  number  of 
variables.     (See  §  74.) 


42  PLANE  ANALYTIC  GEOMETRY         [III,  §  38 

38.    Exceptions.     The   process   of   §  37  cannot  be   applied 
when  the  determinant  of  the  equations  (1)  vanishes,  i.e.  when 


=  0, 


that  is,  when  ajj2  =  ^2^1  • 

For  the  sake  of  simplicity  we  here  assume  that  none  of  the 
four  numbers  a^,  b^,  a^,  6,  is  zero.  If  any  one  of  them  were 
zero,  we  might  solve  the  equation  in  which  it  occurs  to  obtain 
the  value  of  one  of  the  variables.  With  this  assumption,  the 
condition  may  be  written  in  the  form 

02  _  62 
ai      by 
or,  denoting  the  common  value  of  these  quotients  by  m : 

a^  =  ma^f  b^  =  mb^, 
so  that  the  equations  (1)  become 

a^x  +  biy  =  A^i, 
maiX  +  mbiy  =  k^. 
We  must  now  distinguish  two  cases,  according  as  k^  =  m\  or 
^2  ^  mfci.     In  the  former  case,  i.e.  if 

k.^  =  mkx, 
the  second  equation  reduces,  upon  division  by  m,  to  the  first 
equation.  Thus,  the  two  equations  represent  one  and  the 
same  relation  between  x  and  y,  and  are  therefore  not  sufficient 
to  determine  x  and  y  separately.  We  can  assign  to  either 
variable  an  arbitrary  value  and  then  find  a  corresponding 
value  of  the  other  variable.  The  equations  (1)  can  then  be 
said  to  have  an  infinite  number  of  solutions. 
In  the  other  case,  i.e.  if 

the  equations  are  evidently  inconsistent,  and  there  exist  no 
finite  values  of  x  and  y  satisfying  both  equations.  Thus  the 
equations  \x  —  2y  =  2,    2  «  — 12  v/  =  15  are  inconsistent. 


Ill,  §  40]     SIMULTANEOUS  LINEAR  EQUATIONS 


43 


39.  Geometric  Interpretation.  All  these  results  about 
linear  equations  can  be  interpreted  geometrically.  We  have 
seen  (§  30)  that  every  linear  equation  represents  a  straight 
line,  and  (§  35)  that  by  solving  two  such  equations  we  find 
the  coordinates  of  the  point  of  intersection  of  the  two  lines. 
Now  two  lines  in  a  plane  may  either  intersect,  or  coincide,  or 
be  parallel.  In  the  first  case,  they  have  a  single  point  in  com- 
mon ;  in  the  second,  they  have  an  infinite  number  of  points  in 
common ;  in  the  third,  they  have  no  point  in  common.  The 
first  case  is  that  of  §§  36,  37 ;  the  last  two  cases  are  discussed  in 
§  38.  Including  the  case  of  coincident  lines  with  that  of  paral- 
lels, we  may  say  that  the  relation 

is  the  necessary  and  sufficient  condition  of  parallelism  of  the 
two  lines  a^x  -{-  hy  =  h,       a^x  +  b.jy  =  k,. 


=  0 


40.   Elimination.     If  in  the  linear  equations  (1)  of  §  36  the 
constant  terms  k^,  k^  are  both  zero  so  that  they  are 
a^x  +  h^y  =  0, 
a.2X-\-b2y  =  0, 
the  equations  are  called  homogeneous.     Obviously,  two  homo- 
geneous linear  equations  are  always  satisfied  by  the  values 
x=  0,   ^  =  0. 
If  the  determinant  of  ^  the  equations  does  not  vanish,  i.e.  if 

this  solution  is  also  found  from  §  36,  and  it  is  the  only  solution. 
But  if  ai     &i 

it  is  found  as  in  §  38  that  the  equations  have  an  infinite  num- 
ber of  solutions.     Conversely,  if  two  homogeneous  linear  eqiia- 


=0, 


44 


PLANE  ANALYTIC  GEOMETRY  [III,  §  40 


tions  are  satisfied  by  values  of  x  and  y  that  are  not  both  zero,  the 
determinant  of  the  equations  must  vanish.  For,  multiplying  the 
first  equation  by  62;  and  the  second  by  5^,  and  subtracting,  we 

Eliminating  a?  in  a  similar  manner,  we  find 
(ai&2-a2&i)2/=0. 

These  equations  show  that  unless  x  and  y  are  both  zero  we 

must  have 

%     5i 


a^bz  —  a^b^  =  0,    i.e. 


a.2     &2 


=  0. 


This  relation  is  also  the  result  of  eliminating  x  and  y  between 
the  two  equations.  For,  if,  e.g.,  jc  =/=  0  we  may  divide  both 
equations  by  x  and  then  eliminate  y/x  between  the  equations 

ai4-6i^  =  0,      a2  +  &2-=0, 

X  X 

by  multiplying  the  former  by  62)  the  latter  by  b^,  and  subtract- 
ing. The  result  is  again  afiz  —  ajb^^  =  0.  Thus  the  result  of 
eliminating  the  variables  between  two  homogeiieous  linear  equa- 
tions is  the  determinant  of  the  equations  equated  to  zero.  We 
shall  see  later  (§  75)  that  all  the  results  of  the  present  article 
are  true  for  any  number  of  homogeneous  linear  equations. 

Geometrically,  two  homogeneous  linear  equations  of  course 
represent  two  lines  through  the  origin.  The  vanishing  of  the 
determinant  means  that  the  lines  coincide  so  that  they  have  an 
infinite  number  of  points  in  common. 

EXERCISES 

1.   Evaluate  the  determinants  : 


(a) 

2  5 

3  4 

> 

(&) 

7  -3 
4    1 

5 

(c) 

1  a 
-a    1 

' 

id) 

sin^ 
cos  j9 

—  cos  /S 
sin^ 

;  (e) 

1 

cos/3 

30Si8 

1 

;   (/) 

ai  +  a2 
at 

02 

a2  +  a3 

Ill,  §40]    SIMULTANEOUS  LINEAR  EQUATIONS 


45 


2.  Express  x^  +  y-  in  the  form  of  a  determinant  of  the  second  order. 

3.  Verify  that 


a2  +  52      aa'  +  bb' 

a 

b 

2 

and  that 

aa'-\-bb'      a'^-hb'^ 

a'    b' 

> 

a2  +  62  _|_   c2    aa'  +  bh'  +  cc' 
aa'  +  bb'  +  cc'     a'^  +  6'2  _  c'2 

= 

b     c 

b<    c' 

2 

+ 

c      a 
c'     a' 

2      a 
^  a' 

b 
b' 

4.   Verify  that 

\aa' -\-hh'    ac' +bd' 

\a    b\      \a'     b' 

lea 

'  +  db'    cc'  +dd' 

~ 

c 

d 

n 

c'    d' 

(a) 

(d) 


(a) 


(d) 


(a) 


:3x- 6^-8=0, 
[  x-2y  +  l  =  0. 
■4x-2y-7  =  0, 
2x-Sy  +  5  =  0. 


|5x-7y  +  6  =  0, 
l5x-7y+s=0. 


(&) 


(&) 


5.  Find  the  coordinates  of  the  points  of  intersection  of  the  following 
lines  ;  and  check  by  a  sketch  : 

|6x-7y+ll  =  0,  j4x+2y-7zz:0,  •  i2x-by  =  3, 

[3x+2y-12=0.        ^  ^  [3x-8y+4=0.        ^^^  |    x  +  Sy=-l. 
j4x  +  2i/  =  9,  I       3x+2i/=0,  |2.4x+3.l2/=4.5, 

|2x-5y=0.  ^^^  |6x-4i/+4=0.       ^*'^  [    .8x  +  2y=6.2. 

6.  Do  the   following  pairs  of  lines  intersect,  or  are  they  parallel  or 
coincident  ? 

I  3x  +  ?/-6  =  0,  I   3x-5?/=0, 

I   »;  +  i?/-2=0.    ^^^  |]0?/-6x=0. 
f2x-62/-4=0  I    x  +  iy  =  0, 

I    x-3y-2=0.'^^\2x  +  S  y  =  0. 
For  what  values  of  s  do  the  following  pairs  of  lines  become  parallel  ? 
f4x  +  sy-15  =  0,  j   3sx-8?/-13=0,         |7x  -  Uy  +  8  =0, 

|2x-7i/+10  =  0.    ^  ^  {2x-2s^+15=0.  ^^-^  jsx-    2y  +  s=0. 

8.    For  what  values  of  s  do  the  following  pairs  of  lines  coincide  ? 

3x  +  2?/  +  3  =  0,  |3x  +  6y-5  =  0, 

sx  —2y  -\-  s  =0.    ^^M    x  +  sy  — f  =  0. 
Solve  the  following  equations  by  determinants  : 


(«) 


(d) 


{2  U-3V-. 

I? 

jX       y 


25, 
5. 


(&) 


?  =  2, 


Ix     y 


ie) 


I      X2+       ?/2 

[2x2- 3y2 
x2      2/2       3  ' 


i+2i=: 


25, 
5. 


(/) 


f    s  =  16  «2  +  100, 


^^^  {5s  +  <2  =  824 

1  3 

x+y     x—y 

2  ,      5 


x-\-y     x  —  y 


=  -  8, 
=     17. 


46 


PLANE  ANALYTIC  GEOMETRY         [III,  §  41 


PART   XL     EQUATIONS   IN  THREE   UNKNOWNS 
DETERMINANTS   OF   THIRD   ORDER 


41.    Solution  of  Three  Linear  Equations. 

linear  equations  with  three  variables  x,  y,  Zy 


To  solve  three 


(1) 


Kin 


a^x  +  632/  +  Cg^  =  k^, 
in  a  systematic  way,  we  might  first  eliminate  z  between  the 
second  and  third  equations  (by  multiplying  the  second  by  C3, 
the_ third  by  Co,  and  subtracting)  ;  and  then  eliminate  z  between 
the  third  and  first  equations.  We  should  then  have  two  linear 
equations  in  x  and  y,  which  can  be  solved  as  in  §  36.  This 
method  is  long  and  tedious.  But  we  can  find  x  directly  by 
multiplying  the  three  given  equations  respectively  by 


^2^3  ^3^2  — 


63C1  —  biC^  = 


^ 


61C2  —  b^Ci 


and  adding  the  resulting  equations.     For  it  is  readily  verified 
that,  in  the  final  equation,  the  coefficients  of  y  and  z,  viz. 


&i 


+  &2 


^3      C3 


63      C 

are  both  zero 

&2        ^2 


61     Ci 


+  63 


Ci 


+  C2 


+C3 


a 


Cz 


4-a2 


\02       C2IJ 


fcl 


+  A:2 


-fA:3 


^ 


62      C2  I  '       ^  I  &: 

We  find  therefore 

^3       C3 

&2        C2 
^3        C3 

i.e.  if  the  coefficient  of  ic  is  =/=  0, 

^162^3  —  fei&3C2  4-  ^2^3^!  —  kzbiC^  4-  k^biCi  —  A;362Ci 

^1^2<^3  ~  ^1^3C2  +  0^2^361  —  «2&iC3  +  (^5^162  —  O362C1 

Observe  that  the  numerator  is  obtained  from  the  denomina- 
tor by  simply  replacing  every  a  by  the  corresponding  k. 


-[ 


:11 


x  = 


Ill,  §  42]    SIMULTANEOUS  LINEAR  EQUATIONS 


47 


It  can  be  shown  similarly  that  y  is  a,  quotient  with  the  same 
denominator,  and  with  the  numerator  obtained  from  the  de- 
nominator by  replacing  every  b  by  the  corresponding  k ;  and  that 
z  is  a  quotient  with  the  same  denominator  and  the  numerator 
obtained  by  replacing  every  c  by  the  corresponding  k. 

42.  Determinants.  The  common  denominator  of  x,  y,  z  is 
usually  written  in  the  form 

«!      bi      Ci 

(Xo        ^2  2 

*3       C3 


(2) 


and  is  then  called  a  determinant  of  the  third  order.  The  nine 
numbers  ai,  &i,  q,  ag,  62?  ^2?  ^3?  ^3?  Cg  are  called  its  elements ;  the 
horizontal  lines  are  called  the  rows,  the  vertical  lines  the 
columns.  The  diagonal  through  the  first  element  a^  is  called 
the  principal  diagonal  ;  that  through  a^  the  secondary  diagonal. 
By  §  41  we  have 
%     b. 


Ci 

b2 

C2 

h 

C3 

h 

Ci 

C2 

=  % 

h 

+  «2 

h 

+  ^3 

h 

C3 

Ci 

C2 

Gz 

Thus,  a  determinant  of  the  third  order  represents  a  sum  of  six 
terms,  each  term  being  a  product  of 
three   elements   and   containing  one 
and  only  one  element  from  each  row 
and  from  each  column. 

The  most  convenient  method  for 
expanding  a  determinant  of  the 
third  order,  i.e.  for  finding  the  six 
terms  of  which  it  is  the  sum,  is 
indicated  by  the  adjoining  scheme. 


48 


PLANE  ANALYTIC  GEOMETRY         [III,  §  42 


Draw  the  principal  diagonal  and  the  parallels  to  it,  as  in  the 
figure ;  this  gives  the  terms  with  sign  + ;  then  draw  the 
secondary  diagonal  and  the  parallels  to  it ;  this  gives  the  terms 
with  sign  — .     (Compare  §  14.) 

43.  General  Rule.  When  three  linear  equations,  like  (1), 
§  41,  are  given,  the  determinant  (2),  §  42,  of  the  coefficients  of 
X,  y,  z  is  called  the  determinant  of  the  equations.  We  can  now 
state  the  rule  for  solving  the  equations  (1)  when  their  determi- 
nant is  different  from  zero,  by  the  following  formulas  (compare 
§36): 


a;  = 


i.e.  each  of  the  variables  is  the  quotient  of  two  determinants;  the 
denominator  in  each  case  is  the  determinant  of  the  equations,  while 
the  numerator  is  obtained  from  this  common  denominator  by  re- 
placing the  coefficients  of  the  variable  by  the  constant  terms. 

It  will  be  shown  in  solid  analytic  geometry  that  any  linear 
equation  in  x,  y,  z  represents  a  plane.  Hence  by  solving  the 
three  simultaneous  equations  of  §  41  we  find  the  point  (or 
points)  common  to  three  planes. 


h 

&1 

Cl 

(Xl 

h 

Cl 

<h 

h 

h 

h 

b2 

C2 

0^2 

k. 

C2 

a^ 

62 

h 

h 

h 

C3 

,     2/  = 

ag 

h 

C3 

-,2;  = 

q-3 

h 

h 

(h 

h 

<h 

«i 

h. 

Cl 

a. 

h 

Cl 

(h 

h 

C2 

02 

h 

^2 

a^ 

h 

C2 

«3 

h 

C3 

«3 

h 

C3 

«3 

h 

C3 

EXERCISES 


1.   Evaluate  the  determinants  : 


1    2     1 

(a) 

3    1     3 

1     4    1 

• 

0         13 

id) 

4         0    3 

5     -1 

2| 

1 

2 

3 

(&) 

4 

5 

6 

7 

8 

9 

1 

0 

0 

(e) 

X 

1 

0 

y 

z 

1 

(0 


(/) 


-1 

1     2 

7 

0    3 

. 

6 

-4    9 

1 

c     -h 

—  c 

1         a 

h 

—  c 

i| 

Ill,  §44]    SIMULTANEOUS  LINEAR  EQUATIONS 


49 


2.  Show  that 
a-\-b 

b 
0 

3.  Evaluate 

r^  ''•  (a) 


b  0 

b  +  c        c 
c       c  +  d 


\a     b     c     d) 


a 

b 

c 

b 

c 

a 

c 

a 

b 

0 

a 

a 

a 

0 

a 

a 

a 

0 

4.   Solve  by  determinants  : 
2r     =  5, 
(a)  I    X—    y-      z    =0, 
2z     =  1. 


(c) 


(e) 


X—    y  - 
2x-^y- 

x  +  2y-    Sz 
x  +  Sy 
2x-6y-l0z 

1       1       2 


=  7, 
=  4, 
=  -8. 


x^ 


+  8  =  0, 


4  +  A-4+   9  =  0, 


(&) 


(d) 


(/) 


(&) 


Sx  -4y  +1  z   =8, 
2x  +3y   +Qz  =-7, 
X  -    y  =4. 


a;2  + 


3, 


i  +  i-^.+  15 


2X2-      y2  4.3  2;2^        62, 

5  iC2  _  2  2/2  _  3  ^2  ^  _  11. 
2  3 


X—  y     y -z 

4 ^ 

^  +  x     X  —  y 

— +  ^ 

W  —  0        0  +  X 


=  1, 

=  -7, 
=  0. 


44.  Properties  of  Determinants.  —  The  advantages  of  using  de- 
terminants instead  of  the  longer  equivalent  algebraic  expressions  of  the 
usual  kind  will  be  apparent  after  studying  the  principal  properties  of  de- 
terminants and  the  geometrical  applications  that  will  follow. 

(1)  A  determinant  is  zero  whenever  all  the  elements  of  any  roio,  or  all 
those  of  any  column,  are  zero. 

This  follows  from  the  fact  that,  in  the  expanded  form  (§42),  every 
term  contains  one  element  from  each  row  and  one  from  each  column. 

(2)  It  follows,  for  the  same  reason,  that  if  all  elements  of  any  roio  {or 
of  any  column)  have  a  factor  in  common,  this  factor  can  be  taken  out  and 
placed  before  the  determinant;  thus,  e.g., 

ai  mbi  Ci 
a2  mbi  C2 
as    mbs     Cs 


«1 

bi    ci 

a2 

bi      C2 

«3 

&3       C3 

50 


PLANE  ANALYTIC  GEOMETRY  [III,  §  44 


(3)  The  value  of  a  determinant  is  not  changed  by  transposition  ;  i.e. 
by  making  the  columns  the  rows,  and  vice  versa.,  preserving  their  order. 
Thus: 


ai 

bi 

Cl 

ai    a2 

as 

a2 

&2 

C2 

= 

6i    62 

bs 

as 

bz 

Cs 

Cl       C2 

Cs 

for,  by  expanding  the  determinant  on  the  riglit  we  obtain  the  same  six 
terms,  with  the  same  signs,  as  by  expanding  the  determinant  on  the  left. 

(4)  The  interchange  of  any  two  rows  {or  of  any  two  columns)  reverses 
the  sign,  but  does  not  change  the  absolute  value,  of  the  determinant. 

This  also  follows  directly  from  the  expanded  form  of  the  determinant 
(§  42).  For,  the  interchange  of  two  rows  is  equivalent  to  interchanging 
two  subscripts  leaving  the  letters  fixed,  and  this  changes  every  term  with 
the  sign  +  into  a  term  with  the  sign  — ,  and  vice  versa.  The  interchange 
of  two  columns  is  equivalent  to  the  interchange  of  two  letters,  leaving 
the  subscripts  fixed,  which  has  the  same  effect. 

(5)  A  determinant  in  which  the  elements  of  any  row  (column)  are  equal 
to  the  corresponding  elements  of  any  other  row  {column)  is  zero. 

For,  by  (4),  the  sign  of  the  determinant  is  reversed  when  any  two 
rows  (columns)  are  interchanged  ;  but  the  interchange  of  two  equal  rows 
(columns)  cannot  change  the  value  of  the  determinant.  Hence,  denoting 
this  value  by  A,  we  have  in  this  case  —  A  =  A,  i.e.  -4  =  0. 


^ 


EXERCISES 


1.    Show  that 


4 
6 
10 
2.   Evaluate  without  expanding ; 


-3 

4 

2     3    4 

-1 

5 

=  2 

3     1     5 

7 

-9 

5    7     9 

(a) 


2 

-4    3 

7 

14    7 

,         (ft) 

4 

-8    4 

13 
0 
0 


11 

6 

-2 


3.   Without  expanding  show  that 


(c) 


1000 
4 

8 


(a) 


a  b  c 
d  e  f 
a    h    i 


abc 


1      1      1 

be    a    1 

a2    a    1 

dbc    eca    fab 

;      (b)    ca    b    1 

= 

62    b     1 

gbc    hca    iab 

ab    c    1 

c2     c    1 

Ill,  §  45]    SIMULTANEOUS  LINEAR  EQUATIONS 


51 


45.    Expansion  by  Minors.     The  general  type  of  a  determinant  of 
the  third  order  is  often  written  in  the  form 


«11      «12 

«13 

a21      «22 

a23 

asi      0532 

^33 

so  that  the  first  subscript  indicates  the  row,  the  second  the  column  in 
which  the  element  stands.  Any  one  of  the  nine  elements  is  denoted 
by  ttik. 

If  in  a  determinant  of  the  third  order,  both  the  row  and  the  column  in 
which  any  particular  element  anc  stands  be  struck  out,  the  remaining  ele- 
ments form  a  determinant  of  the  second  order,  which  is  called  the  minor 
of  the  element  aik.    Thus  the  minor  of  a23  is 


«ii 

«31 


«12 
«32 


r§45 

J  we  have 

an 

ai2    ai3 

021 

0522       ^23 

«31 

«32       ^33 

«22 

^23 

«32       0533 

«12      ^13 

=  «11 

+  a2i 

+  azi 

«32 

a33 

«12      «13 

a22      ^23 

an 

ai2 

a2i 

0522 

a-ix 

^32 

az\    azz 

an    ai3 

^21       ^23 

=  ai2 

+  ^22 

+  a32 

^21      a23 

a-ix    azz 

an    ai3 

the  right-hand  member  is  called  the  expansion  of  the  determinant  by 
minors  of  the  (elements  of  the)  first  column.  It  should  be  noticed,  how- 
ever, that,  while  the  coeflBcients  of  an  and  asi  in  this  expansion  are  the 
minors  of  these  elements,  the  coefficient  of  a2i  is  minus  the  minor  of  021. 
The  determinant  can  also  be  expanded  by  minors  of  the  second  column  : 

ai3 
a23 
a33 

here  the  coefficients  of  ai2  and  a32  are  minus  the  minors  of  these  elements 
while  the  coefficient  of  a22  is  the  minor  of  a22  itself.  This  expansion  fol- 
lows from  the  previous  one  because  the  value  of  the  determinant  merely 
changes  sign  when  the  first  and  second  columns  are  interchanged. 

Let  the  student  write  out  the  similar  development  in  terms  of  minors 
of  the  third  column. 

As  the  value  of  the  determinant  is  not  changed  by  transposition  (§44 
(3)),  the  detenninant  may  also  be  expanded  by  minors  of  the  elements  of 
any  row. 


62 


PLANE  ANALYTIC   GEOMETRY         [III,  §  46 


46.  Cofactors.  To  sum  up  these  results  briefly,  let  us  denote  by  A 
the  value  of  the  determinant  itself,  and  by  Aih  the  value  of  the  minor  of 
the  element  aa-,  multiplied  hij  (—  1)*+*,  i.e.  the  so-called  cof actor  of  a»fc. 
We  then  have : 

A  =  aixAii  +  a2i^2i  +  «3i^3i , 

=  «12^12  +  «22^22  +  «32^32  , 
=  0513^13  +  «23-423  4-  dZzA  33, 

and  similarly  for  the  expansion  by  minors,  or  rather  cofactors,  of  any  row. 
At  the  same  time  it  should  be  noted  that  if  we  add  the  elements  of  any 
column  (row)  each  multiplied  by  the  cofactors  of  any  other  column  (row), 
the  result  is  always  zero.    Thus  it  is  readily  verified  that 

«11^12  +  «21-422  +  «31-<432  =  0, 
«11^13  +  «21^23  +  «31^33  =  0, 
auAn  +  «22^21  +  «32^31  =  0, 

etc.      This  property  was  used  in  §  4L 


47.  Sum  of  Two  Determinants,  if  all  the  elements  of  any 
column  (or  row)  are  sums,  the  determinant  can  be  resolved  into  a  sum  of 
determinants.  Thus,  if  all  elements  of  the  firs:  column  are  sums  of  two 
terms,  we  find,  expanding  by  minors  of  the  first  column  : 


a2+wi2 
a3+wi3 


(«i  +  wii) 


+  (a2  +  wt2) 


+  (a3+m3) 


= 

ai 

b%    c^ 

&3      C3 

+  052 

bz    C3 
bi    ci 

+  az 

bi    ci 
bi    Ci 

+  mi 

62  C2 

63  C3 

&1      Ci 

bi    Ci 
02    Ca 

ai    61    ci 

Wi      61      Ci 

= 

a2    62    <h 

+ 

ma    62    C2 

' 

Oz 

63     c 

3 

wis    63 

Cs 

Let  the  student  show,  by  interchanging  rows  and  columns,  that  the 
same  property  holds  for  rows. 

As  any  row  (column)  can  be  made  the  first  by  interchanging  it  with  the 
first  and  changing  the  sign  of  the  determinant,  this  decomposition  into 
the  sum  of  two  determinants  is  possible  whenever  every  element  of  any 
one  row  or  column  is  a  sum. 


Ill,  §  47]    SIMULTANEOUS  LINEAR  EQUATIONS 


53 


As  a  particular  case  we  have 


ai-^bi    61    ci 

ai    &i    ci 

61     61     Ci 

«!       61       Ci 

a2  +  &2    62    C2 

= 

a2    &2    C2 

+ 

&2      &2      C2 

= 

(Z2       62      C2 

as  +  bs    63    ^^3 

^3    h    C3 

63      &3      C3 

053       bs      C3 

since  the  second  determinant,  which  has  two  equal  columns,  is  zero  by 
(5),  §  44.  We  conclude  that  the  value  of  a  determinant  is  not  changed 
by  adding  to  each  element  of  any  row  {column)  the  corresponding  element 
of  any  other  row  (column).  Indeed,  owing  to  (2),  §  44,  we  can  add  to 
each  element  of  any  row  (column)  the  corresponding  element  of  any 
other  row  (column)  multiplied  by  one  and  the  same  factor.  This  property 
is  of  great  help  in  reducing  a  given  determinant  to  a  more  simple  form 
and  evaluating  it. 

In  the  case  of  a  numerical  determinant,  it  is  often  best  after  taking  out 
the  common  factors  from  any  row  or  column  to  reduce  two  elements  of 
some  row  or  column  to  zero,  by  addition  or  subtraction.  Thus,  taking 
out  the  factors  2  from  the  third  column  and  3  from  the  second  row, 
we  have 


2 

3 

-14 

2 

3 

-7 

3 

18 

-12 

=  6 

1 

0 

-2 

4 

8 

18 

-4 

8 

9 

subtracting  twice  the  second  row  from  the  first  and  adding  4  times  the 
second  row  to  the  third,  we  find 


A  =  6 


0 

-9 

-3 

-9 

-3 

1 

6 

-2 

=  -«     S2 

J  =18 

0 

32 

1 

=  -622. 


EXERCISES 
1.  Evaluate  the  determinants : 


(a) 


(d) 


1     3      7 

3    5      9 

, 

4    8     16 

6    33      9 

14    21    35 

26    39    4 

12 1 

(ft) 


(e) 


27 

26 

27 

31 

33 

36 

43 

44 

45 

7 

17 

29 

11 

19 

31 

13 

23 

37 

He) 


(/) 


17 

34 

51 

28 

72 

38 

? 

39 

65 

52 

2 

-3 

40 

5 

7 

-10 

3 

-2 

( 

30l 

\b 


54 


PLANE  ANALYTIC  GEOMETRY         [III,  §  47 


2.   8how  that 

b  +  c  a  1 
c  -h  a  b  1 
a+  feci 


(a) 


=  0; 


(&) 


(c) 


(d) 


6i  +  ci 

62  +  C2 

63  +  C3 

a-hb 
a  +  2b 
a  +  Sb 


ci  +  ai 

C2+  a2 
C3+  as 

a  +  46 
a  +  5b 
a  +  66 


1    a'2 

-ff^ 

a3 

-# 

1 

a2 

a3 

1      &2  _  d2      b^ 

-d^ 

= 

1 

62 

63 

1     c2-d-^    c3 

-# 

1 

C2 

C3 

ai  +  &i 

«i 

61     Ci 

a2  +  62 

—  2 

a2 

62     C2 

> 

a3  +  63 

as 

&3      C3 

a  +  76 

a  +  86 

=  0. 

a  +  9b 

48.   Elimination.     Three  hoynogeneous  linear  equations, 


(3) 


a^x  +  61?/  +  CiZ  =  0, 
a^  -f  &2^  +  C22;  =  0, 

«'3«  +  632/  +  C32  =  0, 


are  obviously  satisfied  by  x  =  0,  y  =  0,  z=0.     Can  they  have 
other  solutions? 

Solving  the  equations  by  the  method  of  §  43,  and  denoting 
the  determinant  of  the  equations  for  the  sake  of  brevity  by  A, 
we  find  since  Aij  =  0,  fcg  =  0,  A^a  =  0 : 

Ax=0,  Ay  =  0,  Az  =  0. 

Hence,  if  x,  y,  z  are  not  all  three  zero,  we  must  have  ^  =  0. 
Three  homogeneous  linear  equations  can  therefore  have  solutions 
that  are  not  all  zero  only  if  the  determinant  of  the  equations  is 
equal  to  zero. 

If  X,  for  instance,  is  different  from  zero,  we  can  divide  each 
of  the  three  equations  by  x  and  then  eliminate  y/x  and  z/x  be- 
tween the  three  equations.     The  result  is  ^  =  0,  i.e. 

tti     hi    Ci 

a^    62    C2  =0. 

Ctg  63  Ci 


Ill,  §  49]    SIMULTANEOUS  LINEAR  EQUATIONS  55 

Thus,  the  result  of  eliminating  the  three  variables  betiveen  three 
homogeneous  linear  equations  is  the  determinant  of  the  equations 
equated  to  zero.     (Compare  §  40.) 

Solving  the  first  and  second  equations  for  y/x,  z/Xy  we  obtain 


X 

y 

z 

bi     Ci 

c,     a, 

a,     61 

62     C2 

C2       0^2 

(X2     62 

provided  the  denominators  are  all  different  from  zero. 

With  the  notation  of  §  46,  this  can  be  written  x:y  :  z=Azi  :  ^32  :  ^33- 
If  we  solve  the  third  and  first  or  second  and  third  equations  for  y/x,  z/x^ 
we  find,  respectively,  x  :y:  z  =  A21  :  A22  '■  A23,  or  x  :y  :z  =  An  '•  ^12 :  ^i3- 
Hence,  whenever  ^  =  0,  we  can  find  the  ratios  of  the  variables  unless  all 
the  minors  of  A  are  zero. 

49.  Geometric  Applications.  The  equation  of  a  line 
through  two  points  Pj  {x-^,  2/1)  and  Pg  fe  2/2)  can  be  found  as  fol- 
lows.    The  equation  of  any  line  must  be  of  the  form  (§  30) 

(4)  Ax-irBy+C=^0. 

The  question  is  to  determine  the  coefficients  A,  B,  C,  so  that 
the  line  shall  pass  through  the  points  P^  and  Pg.  If  the  line  is 
to  pass  through  the  point  P^,  the  equation  must  be  satisfied  by 
the  coordinates  x^,  y^  of  this  point,  i.e.  we  must  have 

^1  +  ^^1+0  =  0; 

this  is  the  first  condition  to  be  satisfied  by  the  coefficients.  In 
the  same  way  we  find  the  second  condition 

Ax^-\-  By,^^C=0. 

We  might  calculate  from  these  two  conditions  the  values  of  A/C 
and  B/G  and  then  substitute  these  values  in  the  first  equation. 
But  as  this  means  merely  eliminating  A,  B,  C  between  the 
three  equations,  we  can  obtain  the  result  directly  (§  48)  by 
equating  to  zero  the  determinant  of  the  coefficients  of  A,  B,  C. 


56 


PLANE  ANALYTIC  GEOMETRY         [III,  §  49 


Thus  the  equation  of  the  line  through  two  points  P^,  P^  is 


(5) 


X 

y 

1 

«! 

2/1 

1 

X^ 

2/2 

1 

=  0. 


Observe  that  this  equation  is  evidently  satisfied  if  x^  y  are  re- 
placed either  by  x^,  i/i  or  by  x^,  2/2  (see  (5),  §  44). 

50.  Area  of  a  Triangle.  The  area  ^  of  a  triangle  PiPoP^ 
in  terms  of  the  coordinates  of  its  vertices  Pi(xij  2/i)>  ^^2(^2?  2/2)? 
^3(3^3,2/3)13: 

for,  upon  expanding  this  determinant,  we  find  the  value  given 
before  in  §  14. 

It  will  now  be  seen  that  the  determinant  equation  (5)  of  the 
line  through  two  points  given  in  §  49  merely  expresses  the 
fact  that  any  point  (x,'y)  of  the  line  forms  with  the  given 
points  (xi,  2/1)  and  (x^,  y^  a  triangle  whose  area  is  zero. 


■A  =  \ 


X, 

2/1 

1 

x^ 

2/2 

1 

X, 

Vs 

1 

EXERCISES 

1.  Write  down  the  equation  of  the  line  through  (2,  3),  (—2,  \);  ex- 
pand the  determinant  by  minors  of  the  first  row  ;  determine  the  slope  and 
the  intercepts  ;  sketch  the  line. 

2.  Find  the  equation  of  the  line  through  the  points  :  (3,  —  4)  and 
(0,  2)  ;  (0,  6)  and  (a,  0);  (0,  0)  and  (2,  1). 

3.  Find  the  area  of  the  triangle  whose  vertices  are  the  points  (1,  1), 
(2, -3),  (5,  -8). 

4.  Find  the  area  of  the  quadrilateral  whose  vertices  are  the  points 

(3,  -2),  (4,  -5),  (-3,1),  (0,0). 

6.  If  the  base  of  a  triangle  joins  the  points  (—  1,  2)  and  (4,  3),  on 
what  line  does  the  vertex  lie  if  the  area  of  the  triangle  is  equal  to  6  ? 


Ill,  §50]   SIMULTANEOUS  LINEAR  EQUATIONS 


57 


6.  Find  the  coordinates  of  the  common  vertex  of  the  two  triangles  of 
equal  area  3,  whose  bases  join  the  points  (3,  5),  (6,  —  8)  and  (3,  —  1), 
(2,  2),  respectively. 

7.  Show  that  the  area  of  any  triangle  is  four  times  the  area  of  the 
triangle  formed  by  joining  the  midpoints  of  its  sides. 

8.  Show  that  the  sum  of  the  areas  of  the  triangles  whose  vertices  are 

(a,  d),  (2  6,  c),  (6  c,  f),  and  (Sa,d),  (4&,  e),  (3  c,/)  is  given  by  the 

determinant 

2a    d    1 

Bb     e    1 

4c     /    1 

9.  Show  that  the  lines  joining  the  midpoints  of  the  sides  of  any 
triangle  divide  the  triangle  into  four  equal  triangles. 

10.  Show  that  the  condition  that  the  three  lines  Ax  -{-  By  ■{- C  =  0^ 
A'x  +  B'y  +  C"  =  0,  A"x  +  B"y  +  C"  =  0  meet  at  a  point  is 

ABC 

A'      B'      C    =0. 

A"    B"     C" 

11.  Show  that  the  straight  lines  Sx  +  y  —  l=0,  x—Sy-\-lS  =  0, 
2x— y-{-6=^0  have  a  common  point. 

12.  For  what  values  of  s  do  the  following  lines  meet  in  a  point  : 

4x-Qy-\-s  =  0,  sx-S6y  =  0,x-{-y-l=0? 

13.  Show  that  the  altitudes  of  any  triangle  meet  in  a  point. 

14.  Show  that  the  medians  of  any  triangle  meet  in  a  point. 

15.  Show  that  the  line  through  the  origin  perpendicular  to  the  line 
through  the  points  (a,  0)  and  (0,  b)  meets  the  lines  through  the  points 
(a,  0),  (—  &,  &)  and  (0,  6),  (a,  —  a)  in  a  common  point. 

16.  Show  that  the  distance  of  the  point  Pi(xi,  y{)  from  the  line  joining 
the  points  P2(,X2,  1/2)  and  Pz^xs,  ys)  is 


xi    y\ 

1 

X2    yt 

1 

xz    yz 

1 

y/ixz-x.y^+  (2/3-2^2)2 


CHAPTER   IV 


RELATIONS  BETWEEN   TWO   OR   MORE   LINES 

51.  Angle  between  Two  Lines.  We  shall  understand  by 
the  angle  (I,  V)  =  6  between  two  lines  I  and  I'  the  least  angle 
through  which  I  must  be  turned  coun- 
terclockwise about  the  point  of  inter- 
section to  come  to  coincidence  with  l'. 
This  angle  0  is  equal  to  the  differ- 
ence of  the  slope  angles  a,  a'  (Fig.  27) 
of  the  two  lines.  Thus,  if  a'  >  a,  we 
have  B=  a!  —  a,  since  a'  is  the  exterior 
angle  of  a  triangle,  two  of  whose  interior  angles  are  a  and  6. 
It  follows  that 

tan  a!  —  tan  a    ■ 


Fig.  27 


(1) 


tan  Q  —  tan  {a!  —  a) 


1  -f-  tan  a  tan  a! 
If  the  equations  of  I  and  X  are 

y  =  mx  -\-  b,     y  =  m'x  +  6', 
respectively,  we  have  tan  a  =  m,  tan  a'  =  m' ;  hence 

m'  —  m 


(2) 


tan^  = 


1 4-  mm' 
If  the  equations  of  I  and  I'  are 

^a;  +  52/  -f  C  =0, 

respectively,  we  have  tan  a  =  —  ^1/-B,  tan  a'  =  —  ^'/^'  5  hence 

AB'  -  AB 


(3) 


tan^  = 


AA'+BB^' 

58 


IV,  §  52]  RELATIONS  BETWEEN   LINES  59 

52.  It  follows,  in  particular,  that  the  two  lines  Z  and  Z',  §  51, 
are  parallel  if  and  only  if 

m'  =  m,      or  AB'  -  A'B  =  0 ; 
and  they  are  perpendicular  to  each  other  if  and  only  if 

m'  =  --,  ovAA'  +  BB'=0. 
m 

(Compare  §§  27,  31.)  Hence,  to  write  down  the  equation  of 
a  line  parallel  to  a  given  line,  replace  the  constant  term  by  an 
arbitrary  constant ;  to  write  down  the  equation  of  a  line  per- 
pendicular to  a  given  line,  interchange  the  coefficients  of  x  and 
y,  changing  the  sign  of  one  of  them,  and  replace  the  constant 
term  by  an  arbitrary  constant. 

EXERCISES 

1.  Determine  whether  the  following  pairs  of  lines  are  parallel  or  per- 
pendicular :  3x  +  2?/  —  6  =  0,  2x-3?/  +  4=z0;  5ic  +  3y-6=0, 
10x  +  6y4-2  =  0;2x-|-5y-14=0,  8x-3?/  +  6=0. 

2.  Find  the  point  of  intersection  of  the  Hne  5x  +  8y  +  17=0  with  its 
perpendicular  through  the  origin, 

3.  Find  the  point  of  intersection  of  the  lines  through  the  points  (6,  —2) 
and  (0,  2),  and  (4,  5)  and  (-1,-4). 

4.  Find  the  perpendicular  bisector  of  the  line-segment  joining  the 
point  (3,  4)  to  the  point  of  intersection  of  the  lines  2x  —  y  +  1  =  Q  and 
3  X  4-  2/  -  16  =  0. 

6.  Find  the  lines  through  the  point  of  intersection  of  the  lines  5  x— z/ =0, 
x  +  7i/  —  9  =  0  and  perpendicular  to  them. 

6.  Find  the  area  of  the  triangle  formed  by  the  lines  3  x  +  4  y  =  8, 
6  X  —  5  2/  =  30,  and  x  =  0. 

7.  Find  the  area  of  the  triangle  formed  by  the  lines  x  +  y  —  1  =  0, 
2  X  +  y  +  5  =  0,  and  X  -  2  !/  -  10  =  0. 

8.  Find  the  point  of  intersection  of  the  lines 

(a)  ^  +  f=l,     f  +  I=l. 
ah  ha 

(h)   -  +  |=1,     y  =  mx-\-h- 
a     0 


60  PLANE  ANALYTIC  GEOMETRY  [IV,  §  52 

9.   Find  the  area  of  the  triangle  formed  by  the  lines  y  =  miX  +  6i, 
y  =  m2X  +  &2  and  the  axis  Ox. 

10.  The  vertices  of  atriangle  are  (5,  —  4),  (—  3,  2),  (7,6).     Find  the 
equations  of  the  medians  and  their  point  of  intersection. 

11.  Find  the  angle  between  the  lines  4  x—S  y—G=0  and  x—7  y-\-Q=0. 

12.  Find  the  tangent  of  the  angle  between  the  lines  (a)  4  x—Sy-\-6=0 
and9a;  +  22/-8  =  0;    (b)  3a;  +  6y-ll=0  and  x-\-2y-S  =  0. 

13.  Find  the  two  lines  through  the  point  (6,  10)  inclined  at  45°  to 
the  line  3a;-2?/-12=:0. 

14.  Find  the  lines  through  the  point  (—  3,  7)  such  that  the  tangent  of 
the  angle  between  each  of  these  lines  and  the  line  6.x  —  2i/  +  ll  =  0isJ. 

15.  Show  that  the  angle  between  the  lines  Jtc  +  J5y  +  C  =  0  and 

(A  +  B)x  -{A-  B)y  +  D  =  0  is  45°. 

16.  Find 'the  lines    which    make    an    angle   of    45°    with   the  line 
4x  —  7y  +  6=0  and  bisect  the  portion  of  it  intercepted  by  the  axes. 

17.  The  hypotenuse  of  an  isosceles  right-angled  triangle  lies  on  the  line 
Sx  —  6y-n==0.     The  origin  is  one  vertex  ;  what  are  the  others  ? 

53.  Polar  Equation  of  Line.  The  position  of  a  line  in  the 
plane  is  fully  determined  by  the  length  p  =  ON  (Fig.  28)  of  the 
perpendicular  let  fall  from  the  origin  on 
the  line  and  the  angle  /3  =  xON  made  by 
this  perpendicular  with  the  axis  Ox. 

Then  p  and  /8  are  evidently  the  polm^ 

coordinates  of  the  point  -^  (§  16).     Let 

P  be  any  point  of  the  line  and  OP  =  r, 

xOP—  d>  its  polar  coordinates.     As  the 

Fig  *^8 
projection  of  OP  on   the    perpendicular 

ON  is   equal  to  ON,  and  the  angle  NOP  =  <^  —  ft  we  have 
(4)  rGO{i(<f>  —  p)=p. 

This  is  the  equation  of  the  line  NP  in  polar  coordinates. 


IV,  §54]  TWO  OR  MORE  LINES  61 

54.  Normal  Form.  The  last  equation  can  be  transformed  to 
Cartesian  coordinates  by  expanding  the  cosine  : 

r  cos  <^  cos  P  +  r  sin  <f>sm  p=p 

and  observing  (§  17)  that  r  cos  <l>  =  x,  r  sin  <i>  =  y\  the  equation 
then  becomes 

(5)  ai^cosp+ y8inp=ip. 

This  equation,  which  is  called  the  normal  form  of  the  equation 
of  the  line,  can  be  read  off  directly  from  the  figure ;  it  means 
that  the  sum  of  the  projections  of  x  and  y  on  the  perpendicular 
to  the  line  is  equal  to  the  projection  of  r  (§  20). 

Observe  that  in  the  normal  form  (5)  the  number  p  is  always 
positive,  being  the  distance  of  the  line  from  the  origin,  or  the 
radius  vector  of  the  point  JSf.  Hence  x  cos  ^  -f-  y  sin  ^  is  always 
positive ;  this  also  appears  by  considering  that  x  cos  /3  -\-y  sin  ^ 
is  the  projection  of  the  radius  vector  OP  on  ON,  and  that  this 
radius  vector  makes  with  ON  an  angle  that  cannot  be  greater 
than  a  right  angle. 

The  angle  l3  =  xONiSj  as  a  polar  angle  (§  16),  always  under- 
stood to  be  the  angle  through  which  the  axis  Ox  must  be  turned 
counterclockwise  about  the  origin  to  make  it  coincide  with  ON; 
it  can  therefore  have  any  value  from  0  to  2  tt.  By  drawing  the 
parallel  to  the  line  NP  through  the  origin  it  is  readily  seen 
that,  if  a  is  the  slope  angle  of  the  line  NP,  we  have 

according  as  the  line  lies  on  one  side  of  the  origin  or  the  other, 
angles  differing  by  2  tt  being  regarded  as  equivalent.  Thus,  in 
Fig.  28,  «  =  120°,  /?  =  «+! 7r  =  120°+ 270°  =  390°,  which  is 
equivalent  to  30°.  For  a  parallel  on  the  opposite  side  of  the 
origin  we  should  have  ^  =  «-}- 1  tt  =  120°  +  90°  =  210°. 


62  PLANE  ANALYTIC  GEOMETRY  [IV,  §  55 

55.    Reduction  to  Normal  Form.    The  equation 

Ax-\-By^C=0 
is  in  general  not  of  the  form  (5),  since  in  the  latter  equation 
the  coefficients  of  x  and  y,  being  the  cosine  and  sine  of  an 
angle,  have  the  property  that  the  sum  of  their  squares  is  equal 
to  1,  while  in  the  former  equation  the  sum  of  the  squares  of 
A  and  B  is  in  general  not  equal  to  1.    But  the  general  equation 

Ax-{-By-\-C=0 

can  be  reduced  to  the  normal  form  (5)  by  multiplying  it  by 
a  factor  k  properly  chosen ;  we  know  (§  30)  that  the  equation 

hAx-\-'kBy^'kG=0 

represents  the  same  line  as  does  the  equation  Ax-\-By-\-G=0. 
Now  if  we  select  k  so  that 

kA  =  cos  p,    kB  =  sin  /3,    kC  =  — 1>, 
the  equation   Ax  +  By-\-C=0  reduces  to  the   normal   form 
xQos  p  +  y  sin  p  —  2)  =  0.     The  first  two  conditions  give 

A;M2  +  k'^B'  =  cos2  /3  +  sin^  ^  =  1, 

whence  A;  =  ± 


VA^-hB" 

Since  the  right-hand  member  p  in  the  normal  form  (5)  is  posi- 
tive, the  sign  of  the  square  root  must  be  selected  so  that  kC 
becomes  negative.     We  have  therefore  the  rule  : 
\     To  reduce  the  general  equation  Ax  -\-By-{-C  =0  to  the  normal 

J  form 

\  ajcos^ +  2/ sin/3  —  j9=  0, 


/  divide  by  —  ■\/ A?  +  B^  when   C  is  positive  and  by  -^^A^-\-B^ 

\wjien  C  is  negative. 

Then  the  coefficients  of  x  and  y  will  be  cos  ft  sin  ft  respec- 
tively, and  the  constant  term  will  be  the  distance  p  of  the  line 
from  the  origin. 


IV,  §  56]  TWO  OR  MORE  LINES  63 

Thus,  to  reduce  3a;  +  22/H-5  =  0to  the  normal  form,  divide 
by  _  V3''  +  22  =  -  Vl3 ;  this  gives 

3        .    ^  2  •  5 

cos  B  = ,  sm  «  = Tzn,  —p  = 1=  ; 

VI3  V13  V13 

i.e.  the  normal  form  is 

3  2  5 

7=^ ;=2/  = 


V13        Vl3        V13 

The  perpendicular  to  the  line  from  the  origin  has  the  length 
5/ Vl3  ;  and  as  both  cos  ^  and  sin  ft  are  negative,  this  perpen- 
dicular lies  in  the  third  quadrant.     Draw  the  line. 

Reduce  the  equation  3  a; +  2?/  —  5  =  0  to  the  normal  form. 

^  56.  Distance  of  a  Point  from  a  Line.  If,  in  Fig.  28,  we 
take  instead  of  a  point  P  on  the  line  any  point  Pi  {x^,  2/1) 
not  on  the  line   (Fig.  29),  the  expression  \       ^ 

Xy  cos  P  +  Vi  sin  j8  is  still  the  projection  on 
ON  (produced  if  necessary)  of  the  radius 
vector  OPi.  But  this  projection  OS  differs 
from  the  normal  ON  =  p  to  the  line.  The 
figure  shows  that  the  difference  '  1    \     ,  ~ 

Xy  cos  )8  +  2/1  sin  y8  —  p  =  OaS  —  0N=  N8  fig.  29  ^- 

is  equal  to  the  distance  N^P^  of  the  point  Pj  from  the  line. 

Thus,  to  find  the  distance  of  any  point  Pj  (x^,  2/1)  from  a  line 
whose  equation  is  given  in  the  normal  form 
a;  cos  /8  +  2/  sin  ^  —  p  =  0, 
it  sufiices  to  substitute  in  the  left-hand  member  of  this  equa- 
tion for  X,  y  the  coordinates  x^,  1/1  of  ^^^  point  P^.  The  expression 

iCi  cos  /?  -f  2/1  sin  ^  — p 
then  represents  the  distance  of  P^  from  the  line. 

If  this  expression  is  negative,  the  point  P^  lies  on  the  same 
side  of  the  line  as  does  the  origin ;  if  it  is  positive,  the  point 


64 


PLANE  ANALYTIC  GEOMETRY 


[IV,  §  56 


Pi  lies  on  the  opposite  side  of  the  line.    Any  line  thus  divides  the 
plane  into  two  regions  which  we  may  call  the  positive  and  nega- 
tive regions ;  that  in  which  the  origin  lies  is  the  negative  region. 
To  find  the  distance  of  a  point  Pi  (x^,  y{)  from  a  line  given  in 

the  general  form 

Ax-\-Bi/-i-C=0, 

we  have  only  to  reduce  the  equation  to  the  normal  form  (§  55) 
and  then  apply  the  rule  given  above.    Thus  the  distance  is 
Ax,  +  By,  +  C        ^^       Ax,-^By,-\-C^ 
-  V^-P  +  B"  V^2-f^ 

according  as  C  is  positive  or  negative. 

57.    Bisector  of  an  Angle.     To  find  the  bisectors  of  the 

angles  between  two  lines  given  in  the  normal  form 
x cos  /8  4-  2/  sin  ^—p=zO, 
X  cos  /?'  +  y  «iii  P'  —p'  =  0, 
observe  that  for  any  point  on  either  bisector  its  distances  from 
the  two  lines  must  be  equal  in  absolute  value.      Hence  the 
equations  of  the  bisectors  are 

a;  cos  ^  +  ?/  sin  )8  — p  =  ±  (a;  cos  y8'  +  2/  sin  yS'  — i>'). 
To  distinguish  the  two  bisectors,  ob- 
serve that  for  the  bisector  of  that  pair 
of  vertical  angles  which  contains  the 
origin  (Fig.  30)  the  perpendicular  dis- 
tances are,  in  one  angle  both  positive, 
in  the  other  both  negative ;  hence  the 
plus  sign  gives  this  bisector. 

If  the   equations  of  the   lines   are 
given  in  the  general  form 

Ax  +  By  +  C  =  0,     A'x  -f-  B'y  +  C  =  0, 
first  reduce  the  equations  to  the  normal  form,  and  then  apply 
the  previous  rule. 


Fig.  30 


IV,  §57]  TWO  OR  MORE  LINES  65 

EXERCISES 
1.   Draw  the  lines  represented  by  the  following  equations  : 


(a)  rcos(0-^7r)=6. 

(e)    r  cos  (0  +  f  tt)  =  3. 

(6)  r  cos  (0  -  tt)  =  4. 

(/)  rsin  (0-i7r)  =8. 

(c)  r  cos  0  =  10. 

(g)   rsin  (0  +  |^)  =  7. 

((?)  r  sin  0  =  5. 

(A)  r  cos  (0  -  1  tt)  =  0. 

2.  In  polar  coordinates,  find  the  equations  of  the  lines  :  (a)  parallel  to 
and  at  the  distance  5  from  the  polar  axis  (above  and  below)  ;  (b)  per- 
pendicular to  the  polar  axis  and  at  the  distance  4  from  the  pole  (to  the 
right  and  left)  ;  (c)  inclined  at  an  angle  of  |ir  to  the  polar  axis  and  at 
the  distance  12  from  the  pole. 

3.  Express  in  polar  coordinates  the  sides  of  the  rectangle  OABG  if 
OA  =  6  and  AB  =  9,  OA  being  taken  as  polar  axis. 

4.  What  lines  are  represented  by  (5)  when  p  is  constant,  while  /3 
varies  from  zero  to  2  ir  ?  What  lines  when  p  varies  while  j3  remains  con- 
stant ? 

5.  The  perpendicular  from  the  origin  to  a  line  is  5  units  in  length  and 
makes  an  angle  tan-i  y\-  with  the  axis  Ox.   Find  the  equation  of  the  line. 

6.  Reduce  the  equations  of  Ex.  8,  p.  34,  to  the  normal  form  (5), 

7.  Find  the  equations  of  the  lines  whose  slope  angle  is  150°  and  which 
are  at  the  distance  4  from  the  origin. 

8.  What  is  the  equation  of  the  line  through  the  point  ( —  3,  6)  whose 
perpendicular  from  the  origin  makes  an  angle  of  120^  with  the  axis  Ox  ? 

9.  For  the  line  7a;—  24?/  —  20  =0  find  the  intercepts,  slope,  length 
of  perpendicular  from  the  origin  and  the  sine  and  cosine  of  the  angle 
which  this  perpendicular  makes  with  the  axis  Ox. 

10.  Find  by  means  of  sin  ^3  and  cos  ^  the  quadrants  crossed  by  the  line 
4x  —  5y  =  S. 

11.  Put  the  following  equations  in  the  form  (5)  and  thus  find  p,  sin  /3, 

cos  /3: 

(a)y=mx-\-b.         (b)   - +^  =  1.         (c)3«  =  4y. 
a      b 

12.  Is  the  point  (3,  —  4)  on  the  positive  or  negative  side  of  the  line 
through  the  points  (—  5,  2)  and  (4,  7)  ? 


66  PLANE  ANALYTIC  GEOMETRY  [IV,  §  57 

13.  Is  the  point  (—1,  —  f )  on  the  positive  or  negative  side  of  the  line 
4x-9y-S  =  0? 

14.  Find  by  means  of  an  altitude  and  a  side  the  area  of  the  triangle 
formed  by  the  lines  3a5  +  2i/  +  10  =  0,  4x-3?/+16  =  0,  2cc  +  ?/-4 
=  0.     Check  the  result  with  another  altitude  and  side. 

15.  Find  the  distance  between  the  parallel  lines  (a)  Hx—  6y—  4  =  0 
and  6  X  -  10  y  +  7  =  0  ;  {h)  5  x  +  7  y  +  9  =  0  and  15  x  +  21  y  -  3  =  0. 

16.  What  is  the  length  of  the  perpendicular  from  the  origin  to  the  line 
through  the  point  (—5,  —  4)  whose  slope  angle  is  60"  ? 

17.  What  are  the  equations  of  the  lines  whose  distances  from  the 
origin  are  6  units  each  and  whose  slopes  are  |  ? 

18.  Find  the  points  on  the  axis  Ox  whose  perpendicular  distances  from 
the  line  24  x  ■-  7  ?/  —  16  =  0  are  ±5. 

19.  Find  the  point  equidistant  from  the  points  (4,  —  3)  and  (—2,  1), 
and  at  the  distance  4  from  the  line  3x  —  4?/  —  5  =  0. 

20.  Find  the  line  parallel  to  12  x  —  5?/  —  6  =  0  and  at  the  same  distance 
from  the  origin  ;  farther  from  the  origin  by  a  distance  3. 

21.  Find  the  two  lines  through  the  point  (1,  -y^)  such  that  the  perpen- 
diculars let  fall  from  the  point  (6,  5)  are  of  length  5. 

22.  Find  the  line  perpendicular  to4x  —  7«/  —  10  =  0  which  crosses  the 
axis  Ox  at  a  distance  6  from  the  point  (—  2,  0). 

23.  Find  the  bisectors  of  the  angles  between  the  lines:  (a)  x—y  —4=0 
and  3  X  +  3 y  +  7  =  0  ;  (6)  6x  -  12 y  -  16  =  0  and  24 x  +  7y  +  60  =  0. 

24.  Find  the  bisectors  of  the  angles  of  the  triangle  formed  by  the  lines 
5  X  +  12  y  +  20  =  0,  4  X  —  3  2/  -  6  =  0,  3  X  -  4  y  +  5  =  0  and  the  centerof 
the  circle  inscribed  in  the  triangle. 

25.  Find  the  bisector  of  that  angle  between  the  lines  3  x  —  VS  ?/+ 10=0, 
V2  x  +  y  —  6  =  0in  which  the  origin  lies. 

26.  If  two  lines  are  given  in  the  normal  form,  what  is  represented  by 
their  sum  and  what  by  their  difference  ? 

27.  Show  that  the  angle  between  the  lines  x  +  y  =  0  and  x  —  y  =  0  is 
90°  whether  the  axes  are  rectangular  or  oblique. 


IV,  §  58]  TWO  OR  MORE  LINES  67 

58.  Pencils  of  Lines.  All  lines  through  one  and  the  same 
point  are  said  to  form  a  pencil;  the  point  is  called  the  center  of 
the  pencil.     If 

^^  \A'x  +  B'y-^C'=:0 

are  any  two  differeijjt-iilies  of  a  pencil,  the  equation 

(7)  Ax-\-By+C+k(A'x-hB'y-]-C')=0, 

where  k  is  any  constant,  represents  a  line  of  the  pencil.  For, 
the  equation  (7)  is  of  the  first  degree  in  x  and  y,  and  the  coeffi- 
cients of  X  and  y  cannot  be  both  zero,  since  this  would  mean 
that  the  lines  (6)  are  parallel.  Moreover,  the  line  (7)  passes 
through  the  center  of  the  pencil  (6)  because  the  coordinates  of 
the  point  that  satisfies  each  of  the  equations  (6)  also  satisfy 
the  equation  (7). 

All  lines  parallel  to  the  same  direction  are  said  to  form  a 
pencil  of  parallels.  It  is  readily  seen  that  if  the  lines  (6)  are 
parallel,  the  equation  (7)  represents  a  line  parallel  to  them. 

EXERCISES 

1.  Find  the  line:  (a)  through  the  point  of  intersection  of  the  lines 
4  ic  —  7  y  +  5  =0,  6aj  + 11  y  —  7=0  and  the  origin  ;  (6)  through  the 
point  of  intersection  of  the  lines  4a;  —  2y  —  3  =  0,  x^-y  —  5  =  0  and 
the  point  (—2,  3)  ;  (c)  through  the  p^nt  of  intersection  of  the  lines 
4ic  —  5?/  +  6  =  0,  y  —  x  —  S  =  0,  of  slope 3  ;  (d)  through  the  intersection 
of5x  —  62/4-10  =  0,  2x  +  3y—  12  =  0,  perpendicular  to  4  y  +  a;  =  0. 

2.  Find  the  line  of  the  pencil  x—  5  =  0,  y  -\-2  =  0  that  is  inclined  to 
the  axis  Ox  at  30°. 

3.  Determine  the  constant  b  of  the  line  y  =  3x+  b  so  that  this  line 
shall  belong  to  the  pencil  Sx  —  iy  +  6  =  0,  x  =  6. 

4.  Find  the  line  joining  the  centers  of  the  pencils  x  —  Sy  =  12, 
5x—  2y  =  1  and  x-{-y  =  6,  4tx  —  5y  =  S. 

5.  Find  the  line  of  the  pencil  4x-5y-12  =  0,  3a;  +  22/-16=0 
that  makes  equal  intercepts  on  the  axes. 


68  PLANE  ANALYTIC  GEOMETRY  [IV,  §  59 

69.   Non-linear  Equations  representing  Lines.    When  two 

lines  are  given,  say 

Ax-{-By-\-C=0, 

then  the  equation 

{Ax  -f  JB2/  +  C){A'x  +  By  +  0')  =  0, 

obtained  by  multiplying  the  left-hand  members  (the  right-hand 
members  being  reduced  to  zero)  is  satisfied  by  all  the  points 
of  the  first  given  line  as  well  as  all  the  points  of  the  second 
given  line,  and  by  no  other  points. 

The  product  equation  which  is  of  the  second  degree  is  there- 
fore said  to  represent  the  two  given  lines.  Similarly,  by  equat- 
ing to  zero  the  product  of  the  left-hand  members  of  the  equations 
of  three  or  more  straight  lines  (whose  right-hand  members  are 
zero)  we  find  a  single  equation  representing  all  these  lines. 
An  equation  of  the  7ith  degree  may  therefore  represent  n 
straight  lines,  viz.  when  its  left-hand  member  (the  right-hand 
member  being  zero)  can  be  resolved  into  n  linear  factors,  with 
real  coefficients. 

EXERCISES 

1.  Find  the  common  equation  of  the  two  axes  of  coordinates. 

2.  Show  that  n  lines  through  the  origin  are  represented  by  a  homo- 
geneous equation  (i.e.  one  in  \^ich  all  terms  are  of  the  same  degree  in 
X  and  y)  of  the  nth  degree. 

3.  Draw  the  lines  represented  by  the  following  equations : 
(a)  (x  -a)(y-b)=  0.  (/)  xy  -  ax  =  0. 

(6)  3x^-xy-4y'^  =  0.  (g)  y^  -  ^y^  ■¥  Qy  =  0. 

(c)  rK2  _  9  1/2  =  0.  (h)  yfiy-xy  =  0. 

{d)  ax"^  +  6^2  =  0.  (0   y^-Q  xy"^  +  11  x^y  -  6  a;^  =  0. 

(e)  a:2  -  iK  -  12  =  0. 

4.  What  relation  must  hold  between  a,  h,  b,  if  the  lines  represented 
by  ax^  -\-2hxy  +  by^  =  0  are  to  be  real  and  distinct,  coincident,  imag- 
inary ?  • 


IV,  §  59]  TWO  OR  MORE  LINES  69 

MISCELLANEOUS  EXERCISES 

1.  Find  the  angle  between  the  lines  represented  by  the  equation 
ayi^  +  2  hxy  +  hy'^  —  0.  What  is  the  condition  for  these  lines  to  be  per- 
pendicular ?  coincident  ? 

2.  Reduce  the  general  equation  Ax  -\-  By  -{-  C  =  0  to  the  normal 
form  xoos  p  +  y  sin  j3  =  p  by  considering  that,  if  both  equations  represent 
the  same  line,  the  intercepts  must  be  the  same. 

3.  Find  the  line  through  (xi ,  yi)  making  equal  intercepts  on  the  axes. 

4.  Find  the  area  of  the  triangle  formed  by  the  hues  y  =  miX  +  6i , 
y  =  m2X  -\-  b2  >,  y  =  b. 

5.  What  does  the  equation  0  =  const,  represent  in  polar  coordinates  ? 

6.  Find  the  polar  equation  of  the  line  through  (6,  v)  and  (4,  |  nr). 

7.  Derive  the  determinant  expression  for  the  area  of  a  triangle  (§14) 
by  multipljdng  one  side  by  half  the  altitude. 

8.  The  weights  lo,  W  being  suspended  at  distances  d,  Z),  respectively, 
from  the  fulcrum  of  a  lever,  we  have  by  the  law  of  the  Jever  WD  =  icd. 
If  the  weights  are  shifted  along  the  lever,  then  to  every  value  of  d  cor- 
responds a  definite  value  of  D ;  i.e.  i>  is  a  function  of  d.  Represent  this 
function  graphically  ;  interpret  the  part  of  the  line  in  the  third  quadrant. 

9.  A  train,  after  leaving  the  station  yl,  attains  in  the  first  6  minutes, 
li  miles  from  A,  the  speed  of  30  miles  per  hour  with  which  it  goes  on. 
How  far  from  A  will  it  be  50  minutes  after  starting?  (Compare  Ex- 
ample 4,  §  29.)     Illustrate  graphically,  taking  s  in  miles,  t  in  minutes. 

10.  A  train  leaves  Petroit  at  8  hr.  25  m.  a.m.  and  reaches  Chicago  at 
4  hr.  5  m.  p.m.  ;  another  train  leaves  Chicago  at  10  hr.  30  m.  a.m.  and 
arrives  in  Detroit  at  5  hr.  30  m.  p.m.  The  distance  is  284  miles.  Regard- 
ing the  motion  as  uniform  and  neglecting  the  stops,  find  graphically  and 
analytically  where  and  when  the  trains  meet.  If  the  scale  of  distances 
(in  miles)  be  taken  1/20  of  the  scale  of  times  (in  hours),  how  can  the 
velocities  be  found  from  the  slopes  ? 

11.  A  stone  is  dropped  from  a  balloon  ascending  vertically  at  the  rate 
of  24  ft. /sec;  express  the  velocity  as  a  function  of  the  time  (Example  5, 
§  29) .     What  is  the  velocity  after  4  sec.  ? 

12.  How  long  will  a  ball  rise  if  thrown  vertically  upward  with  an 
initial  velocity  of  100  ft.  /sec.  ? 


CHAPTER   V 

PERMUTATIONS   AND   COMBINATIONS.     DETERMI- 
NANTS  OF  ANY    ORDER 

60.  Introduction.  In  using  determinants  of  the  second  and 
third  order  we  have  seen  how  advantageous  it  is  to  arrange 
conveniently  the  symbols  of  an  algebraic  expression.  Before 
proceeding  to  the  study  of  the  general  determinant  of  the  wth 
order,  we  must  discuss  very  briefly  that  branch  of  algebra 
which  is  concerned  with  the  theory  of  arrangements  and 
changes  of  arrangement  (permutations  and  combinations). 
The  results  are  important  not  only  for  determinants,  but  are 
used  very  often,  even  in  the  common  affairs  of  life  ;  they  form, 
moreover,  the  basis  of  the  theory  of  "  choice  and  chance,"  or  of 
probabilities. 

The  "  things  "  to  be  arranged  or  combined  need  not  be  num- 
bers (as  they  are  in  a  determinant),  but  may  be  any  what- 
ever, provided  they  are,  and  remain,  clearly  distinguishable 
from  each  other ;  we  shall  call  them  elements  and  designate 
them  by  letters  a,  h,  c,  etc. 

61.  Permutations.  Any  two  elements,  a  and  6,  can  obvi- 
ously be  arranged  in  a  row  in  2  ways : 

ah,     ha. 

Three  elements  a,  5,  c  can  be  arranged  in  a  row  in  6,  and  only 

6,  ways: 

ahc     hac     cab 

acb    hca    cha 
The  question  arises:    in  how  many  ways  can  ^i  elements  be 
arranged  in  a  row  ? 

70 


V,  §62]     PERMUTATIONS  AND  COMBINATIONS  71 

Any  arrangement  of  n  elements  in  a  row  is  called  a  permu- 
tation. It  is  found  by  trial  that  the  number  of  permutations  of 
n  elements  increases  very  rapidly  with  their  number  n.  Thus 
for  4  elements  it  is  24,  for  5  elements  120.  It  will  be  shown 
that  for  n  elements  the  number  of  permutations  zs  1  •  2  •  3  •  •  •  n. 
This  expression,  the  product  of  the  first  n  positive  integers,  is 
briefly  designated  by  n !,  or  \n,  and  is  called  factorial  n : 

n!  =  l -2  .3...%. 

If  we  denote  by  P„  the  number  of  permutations  of  n  ele- 
ments our  proposition  is 

62.  Mathematical  Induction.  The  proof  of  the  proposition 
that  P^  =  nl  is  obtained  by  an  important  method  of  reasoning 
called  mathematical  induction. 

By  actual  trial  we  can  readily  find  that  P^  =  1,  Pg  =  2, 
Pg  =  6,  and  with  sufficient  patience  we  might  even  ascertain 
that  Pe  =  720.  But  to  prove  the  general  proposition  that 
P^  =  7i!  we  must  look  into  the  method  by  which  in  the 
particular  cases  we  make  sure  that  we  have  found  all  the  pos- 
sible permutations.  This  method  consists  in  proceeding  step 
by  step : 

Seeing  that  2  elements  have  2  permutations,  we  form  the 
permutations  of  3  elements  by  taking  each  of  the  3  elements 
and  associating  with  it  the  2  permutations  of  the  remaining 
two ;  we  thus  find  that  Pg  =  3  •  2  =  6. 

Similarly,  to  form  the  permutations  of  4  elements  we  asso- 
ciate each  of  the  4  with  the  6  permutations  of  the  remaining  3  ; 
this  gives  P4  =  4  .3!  =  4! 

This  leads  us  to  expect  that  P^  =  nl  The  actual  proof  rests 
on  two  facts :  (a)  the  special  fact,  found  by  actual  trial,  that 


72  PLANE  ANALYTIC  GEOMETRY  [V,  §62 

e.g.  P2  =  2  ! ;  (6)  the  general  law  that  the  number  of  permuta- 
tions of  n-\-l  elements  is  found  by  associating  each  of  the 
n  -f  1  elements  with  the  P„  permutations  of  the  remaining  71, 
i.e.  that 

-p„+.=(»+i)^„- 

Knowing  from  (a)  that  P2  =  2  !  we  find  from  this  formula  that 
P3  =  3  .  P2  =  3  .  2 !  =  3 ! ;  in  the  same  way  that  P4  =  4  .  3 !  =  4 ! 
etc. 

Notice  that  mathematical  induction  is  not  merely  a  method 
of  trial  and  experiment.  It  requires  that  we  should  know  not 
only  one  special  case  of  the  general  formula  to  be  proved,  but 
also  the  law  by  which  we  can  proceed  from  every  special  case  to 
the  next,  i.e.  from  n  to  ?i  +  1  whatever  the  value  of  n.  This  law 
is  a  result,  not  of  trial  or  induction,  but  of  deductive  reasoning. 
In  our  case  it  is  expressed  by  the  formula  P„+i  =  (n  -{-  1)P„. 
The  method  of  mathematical  induction  is  therefore  often  called 
reasoning  from  n  to  n-\-l. 

63.  Permutations  by  Groups.  A  somewhat  more  general 
problem  in  permutations  is  suggested  by  the  following  exam- 
ple: In  an  office  there  are  two  vacancies,  one  at  $1000,  the 
other  at  $800.  There  are  5  applicants  for  either  of  the  2 
positions ;  in  how  many  ways  can  the  positions  be  filled  ? 

The  first  vacancy  can  be  filled  in  5  ways,  and  then  the  sec- 
ond can  still  be  filled  in  4  ways ;  hence  there  are  5  •  4  =  20 
ways.  Denoting  the  applicants  by  a,  &,  c,  d,  e  the  20  possi- 
bilities are : 


ab 

ac 

ad 

ae 

ha 

be 

bd 

be 

ca 

cb 

cd 

ce 

da 

db 

do 

de 

ea 

eb 

ec 

ed 

V,  §64]     PERMUTATIONS  AND  COMBINATIONS  73 

The  general  problem  here  suggested  is  that  of  finding  the 
number  of  permutations  of  n  elements  k  at  a  time,  where  Tc  <n. 

Each  permutation  here  contains  k  elements ;  and  we  have  to 
fill  the  k  places  in  all  possible  ways  from  the  n  given  elements. 
The  first  place  can  be  filled  in  n  ways.  The  second  can  then  be 
filled  in  ?i  —  1  ways ;  hence  the  first  and  second  places  can  be 
filled  in  n(ii  —  1)  ways.  The  third  place,  when  the  first  two  are 
filled,  can  still  be  filled  in  n  —  2  ways,  so  that  the  first  three 
places  can  be  filled  in  n{n  —  V)(ji  —  2)  ways.  Proceeding  in  this 
way  we  find  that  the  k  places  can  be  filled  in  ti  (n  —  1)  (n  —  2)  ••• 
(n  -~k-{-l)  ways. 

Thus  the  number  of  permutations  of  n  elements,  A;  at  a  time, 
which  is  denoted  by  „P;t,  is 

„P,  =  n{n  -  l){n  _  2)  ...  (n  -  fc  + 1). 

Notice  that  in  ^P^  there  are  as  many  factors  as  places  to  be 
filled,  viz.  k ;  the  first  factor  being  n,  the  second  n  —  1,  etc.,  the 
A:th  will  be  n  —  {k --  1)  =  n  —  k  -\-l. 

lik^nwQ  have  the  case  of  §  61 ;  i.e.  „P„  ==  P^. 

As  ?i!  =  n(?i  — 1)  ...  (n-'k-\-l)  •  {n  —  k){7i  —  k  —  l)  ..-2.1 
=  n{n  —  1 )  '"  {n  —  k-\-l)  •  {n  —  k)\,  the  expression  for  „P^  can 
also  be  written  in  the  form 

p  _      n! 


{n-k)\ 

64.  Combinations.  If,  in  the  problem  of  §  63,  the  2 
vacancies  to  be  filled  are  positions  of  the  same  rank  (as  to 
salary,  qualifications  required,  etc.),  the  answer  will  be  differ- 
ent. We  have  now  merely  to  select  in  all  possible  ways  2  out 
of  5  applicants,  the  arrangements  ah  and  ha,  ac  and  ca,  etc., 
being  now  equivalent.  Therefore  the  answer  is  now  20  divided 
by  2,  i.e.  10,  as  can  readily  be  verified  directly :  ah,  ac,  ad,  oe, 
be,  bdj  be,  cd,  ce,  de. 


74  PLANE  ANALYTIC  GEOMETRY  [V,  §  64 

If  there  were  3  vacancies,  the  number  of  ways  of  filling 
them  from  4  applicants,  when  the  positions  are  different,  is 
4P3  =  4  .  3  •  2  =  24 ;  but  when  the  positions  are  alike,  the 
number  is  24  divided  by  the  number  of  permutations  of  3 
things,  i.e.  24/6  =  4. 

A  set  of  k  elements  selected  out  of  n,  when  the  arrangement 
of  the  k  elements  in  each  set  is  indifferent,  is  called  a  combina- 
tion. The  number  of  combinations  of  k  elements  that  can  be 
selected  from  n  elements  is  denoted  by  ^C^ ;  to  find  this  num- 
ber we  may  first  form  the  number  ^P^  of  permutations  of  n 
elements  A;  at  a  time,  and  then  divide  by  the  number  Pj^=ik\ 
of  permutations  of  k  elements.     Thus 

p  _n{n  —  l)  ••'  {n  —  k-\-l)  _         n\ 
"   *~  1.2  ...A;  ~k\{n-k)\' 

The  number  of  combinations  of  n  elements  that  can  be 
selected  from  n  elements  is  clearly  1 ;  indeed,  for  A:  =  n  our 
first  expression  gives  ^(7„  =  1. 

EXERCISES 

1.  Find  the  value  of  n  if  ^  * 

(a)   ^  =  5.  ^  ,    {hy  ^■=  20.  (c)  p.  =  40320. 

2.  Show  that 

(a)    nGk  =  nGn-lc>        (&)    nCk"^ nCk-^=  n+lCk.        (c)    A;n+lC*=  (n  + 1)„C*-1. 

3.   Prove  by  mathematical  Induction  that : 

(a)  1  +  2  +  3  +  -  +  n  =  I  n(n  +  1). 

(5)  12  +  22  +  3'-'+  •••  +n2  =  ^n(n+l)(2n  +  l). 

(c)  13  +  23  +  33+  ...  +n3=[^n(n  +  l)]2  =  (l  +2  +  3+  •.•  +  n)2. 

{d)  1  +  3  +  5  +  ...  +(2  n  -  1)=  n2. 

(c)  2  +  4  +  6  +  ...  +  2  n  =  n{n  +  1). 

(/)  1.2  +  2.3  +  3.4+  ...  +n(7i  +  l)=in(w  +  l)(n  +  2). 

(9')  T~^  +  ^7—^  +  ^7— ;  +  •••  + 


1.22.33.4  n{n  +  \)      n  +  \ 


V,  §65]    PERMUTATIONS  AND  COMBINATIONS  75 

4.  A  pile  of  shot  forms  a  pyramid  with  n  shot  on  a  side  at  the  base. 
How  many  shot  in  the  pile  if  the  base  is  a  square  ?  an  equilateral  triangle  ? 

6.  Three  football  teams  plan  a  series  of  games  so  that  each  team  will 
play  the  other  two  teams  4  times.     How  many  games  in  the  schedule  ? 

6.  In  how  many  ways  can  a  committee  of  3  freshmen  and  2  sopho- 
mores be  chosen  from  8  freshmen  and  5  sophomores  ? 

7.  In  how  many  ways  can  the  letters  of  the  word  equal  be  arranged 
in  a  row  four  letters  at  a  time  ? 

8.  From  the  26  letters  of  the  alphabet,  in  how  many  ways  can  four 
different  letters,  one  of  which  is  d,  be  arranged  in  a  row  ? 

9.  How  many  numbers  of  three  digits  each  can  be  formed  with  1, 
2,  3,  4,  5,  no  digit  being  repeated  ?  How  many  of  these  numbers  are 
even  ?  odd  ? 

10.  From  a  company  of  60  men,  how  many  guards  of  4  men  can  be 
formed?  How  many  times  will  one  man  (A)  serve  ?  How  many  times 
will  A  and  B  serve  together  ? 

11.  Which  is  the  largest  of  the  numbers  „(7i,  „C2,  nOs,  •••  „0„_i, 
when  n  is  even  ?  odd  ? 

12.  How  many  straight  lines  are  determined  by  12  points,  no  3  of 
which  are  in  a  line  ? 

13.  How  many  triangles  are  determined  by  10  points,  no  3  of  which 
are  in  a  line  ? 

65.  Inversions  in  Permutations.  When  n  elements  ai,  ag, 
as,  •••  a„,  distmguished  by  their  subscripts,  are  given,  their  arrange- 
ment, with  the  subscripts  in  the  natural  order  of  increasing  numbers, 

is  called  the  principal  permutation.  In  every  other  permutation  of  these 
elements  it  will  occur  that  lower  subscripts  are  preceded  by  higher  ones. 
Every  such  occurrence  is  called  an  inversion.  Any  permutation  is  called 
even  or  odd  according  as  the  number  of  inversions  occurring  in  it  is  even 
or  odd.  The  principal  permutation,  which  has  no  inversion,  is  classed  as 
even.     To  count  the  number  of  inversions  in  a  given  permutation,  take 


76  PLANE  ANALYTIC  GEOMETRY  [V,  §  65 

each  subscript  in  order  and  see  by  how  many  higher  subscripts  it  is 
preceded.    Thus,  in  the  permutation 

the  subscript  1  is  preceded  by  the  higher  subscripts  2,  3,  5  (3  inversions); 
2  and  3  are  preceded  by  no  higher  subscripts ;  4  is  preceded  by  5  (1  in- 
version) ;  5,  6,  7  are  not  preceded  by  any  higher  subscripts.  Hence  there 
are  3  -f  1  =  4  inversions,  and  the  permutation  is  even.     The  permutation 

of  the  same  elements  has  3  +  3  +  2  +  3  +  2  =  13  inversions,  and  is,  there- 
fore, odd. 

66.  If  in  a  permutation  any  two  adjacent  elements  are  interchanged^ 
the  number  of  inversions  is  changed  by  1 ;  hence  the  class  to  which  the 
permutation  belongs  is  changed  (from  even  to  odd  or  from  odd  to  even). 

Let  the  two  adjacent  elements  be  ah,  au  and  suppose  that  h<.k. 
Two  cases  arise  according  as  the  original  arrangement  is  ahak  or  ata/,. 

(a)  If  the  original  arrangement  is  a^a^:  i  the  new  arrangement  is  a^rt/, ; 
as  A  <  k^  and  as  all  other  elements  of  the  permutation  remain  unchanged, 
the  number  of  inversions  is  increased  by  1. 

(Z>)  If  the  original  arrangement  is  akCih ,  the  new  arrangement  is  anak 
so  that  the  number  of  inversions  is  diminished  by  1." 

67.  If  in  a  permutation  any  two  elements  lohatever  be  interchanged,  the 
number  of  inversions  is  changed  by  an  odd  number,  and  hence  the  class 
of  the  permutation  is  changed. 

Eor,  the  interchange  of  any  two  elements  a^,  ai,  can  be  effected  by  a 
number  of  successive  interchanges  of  adjacent  elements.  If  there  are  m 
elements  between  an  and  ak,  we  have  only  to  interchange  a^  with  the  first 
of  these  elements,  then  with  the  next,  and  so  on,  finally  with  ak,  and 
then  ak  with  the  last  of  the  m  elements,  with  the  next  to  the  last,  and  so 
on  ;  thus  in  all  wi  +  1  +  m  =  2  wi  +  1  interchanges  of  adjacent  elements 
are  required,  i.e.  an  odd  number. 

68.  Of  the  n  !  permutations  of  n  elements  just  one  half  are  even,  the 
other  half  are  odd. 

This  follows  by  observing  that  if  in  each  of  the  n  !  permutations  we 
interchange  any  two  elements,  the  same  in  all,  every  even  permutation 


V,  §69]         DETERMINANTS  OF  ANY  ORDER 


77 


becomes  odd  and  every  odd  permutation  becomes  even,  and  no  two  differ- 
ent permutations  are  changed  into  the  same  permutation.  After  this 
interchange  we  must  have  exactly  the  same  n !  permutations  as  before. 
Hence  the  number  of  even  permutations  must  equal  that  of  the  odd 
permutations. 

The  propositions  about  inversions  are  important  for  the  theory  of  de- 
terminants of  the  nth  order  to  which  we  now  proceed. 

69.   General  Definition  of  Determinant.    When  n^  numbers  are 

given  (e.g.  the  coefficients  of  the  variables  in  n  linear  equations),  arranged 
in  a  square  array,  we  denote  by  the  symbol 

«ii  •••  «i, 

and  call  determinant  of  the  nth  order  the  algebraic  sum  of  the  n  !  terms 
obtained  as  follows  :  the  first  term  is  the  product  of  the  n  numbers  in  the 
principal  diagonal  aiia22«33  •••  «„n  ;  the  other  terms  are  derived  from  this 
term  by  permuting  in  all  possible  ways  either  the  second  subscripts  or 
the  first  subscripts,  and  multiplying  each  term  by  +  1  or  —  1  according 
as  it  is  an  even  or  odd  permutation  (i.e.  contains  an  even  or  odd  number 
of  inversions) . 

It  follows  at  once  that  every  term  contains  n  factors,  viz.  one  and  only 
one  from  each  row,  and  one  and  only  one  from  each  column. 

It  is  readily  seen  that  this  definition  gives  in  the  case  of  determinants 
of  the  second  and  third  order  the  expressions  previously  used  as  defining 
such  determinants.    For  a  determinant  of  the  fourth  order, 


«11 

an 

ai3 

au 

an 

«22 

^28 

^24 

azi 

az2 

ass 

a34 

an 

«42 

^43 

au 

we  obtain  the  1  •  2  •  3  •  4  =  24  terms  from  the  principal  diagonal  term 

ana^asiau 
by  forming  all  the  permutations,  say  of  the  second  subscripts  1,  2,  3,  4 
and  assigning  the  +  or  —  sign  according  to  the  number  of  inversions.     If 
these  permutations  are  derived  by  successive  interchanges  of  two  sub- 
scripts the  terms  will  have  alternately  the  +  and  —  sign. 


78  PLANE  ANALYTIC  GEOMETRY  [V,  §  70 

70.  The  properties  of  the  determinant  of  the  nth  order  are  essentially 
the  same  as  those  of  the  determinant  of  the  third  order  (§§  44-47). 

(1)  The  determinant  is  zero  whenever  all  the  elements  of  any  row,  or 
all  those  of  any  column,  are  zero. 

For,  every  term  contains  one  element  from  each  row  and  one  from 
each  column. 

(2)  It  follows  from  the  same  observation  that  if  all  elements  of  any 
row  {or  of  any  column)  have  a  factor  in  common,  this  factor  can  be  taken 
out  and  placed  before  the  determinant. 

(8)  The  value  of  a  determinant  is  not  changed  by  transposition;  i.e. 
by  making  the  columns  the  rows,  and  vice  versa,  preserving  their 
order. 

For,  this  merely  interchanges  the  subscripts  of  every  element,  i.e.  the 
first  series  of  subscripts  becomes  the  second  series,  and  vice  versa. 

Hence  any  property  proved  for  rows  is  also  true  for  columns. 

(4)  The  interchange  of  any  two  rows  (columns)  reverses  the  sign  of 
the  determinant. 

For,  the  interchange  of  any  two  rows  gives  an  odd  number  of  inver- 
sions to  the  first  series  of  subscripts  in  the  principal  diagonal  (§  67),  and 
does  not  alter  the  second  series.  Hence  the  signs  of  all  the  terms  are 
reversed. 

CoR.  1.  A  determinant  in  which  the  elements  of  any  row  {column) 
are  equal  to  the  corresponding  elements  of  any  other  row  {column)  is  zero. 

For,  the  sign  of  the  determinant  is  reversed  when  any  two  rows 
(columns)  are  interchanged  ;  but  the  interchange  of  two  equal  rows 
(columns)  cannot  change  the  value  of  the  determinant.  Hence,  denot- 
ing this  value  by  A,  we  have  in  this  case  —  A  =  A,  i.e.  A  =  0. 

(5)  If  all  the  elements  of  any  row  (column)  are  sums  of  two  terms,  the 
determinant  can  be  resolved  into  a  sum  of  two  determinants. 

For,  in  the  expansion  of  the  determinant  every  term  contains  one  bi- 
nomial factor  ;  therefore  it  can  be  resolved  into  two  terms.  See  §  47  for 
an  illustration. 

By  means  of  this  property,  prove  the  following  corollaries : 

CoR.  1.  If  all  the  elements  of  any  row  (column)  are  algebraic 
sums  of  any  number  of  terms,  the  determinant  can  be  resolved  into  a 
corresponding  number  of  determinants. 

Cor.  2.     The  value  of  a  determinant  is  not  changed  by  adding  to  the 


V,  §  70]         DETERMINANTS  OF  ANY  ORDER 


79 


elements  of  any  row  (column)  those  of  any  other  row  {column)  multiplied 
by  any  common  factor. 

This  corollary  furnishes  a  method  (see  §  72)  by  which  all  the  elements 
but  one  of  any  row  (column)  can  be  reduced  to  zero. 


EXERCISES 

1.  How  many  inversions  are  there  in  the  following  permutations  ? 
(a)  aia^aza^aia^a^.        (b)  a7a6aiasa2aiaB.        (c)  a7a6«5«4a3a2«i- 

2.  In  the  expansion  of  the  determinant  below,  what  sign  must  be 
placed  before  the  terms  celn,  agjp  ? 


3.    Show  that 


a 

b 

c 

d 

e 

f 

g 

h 

i 

J 

k 

I 

m 

n 

0 

P 

aix  +  biy  +  Ciz  ai  &i  Ci 

a^X  +  biy  +  C2Z  «2  &2  C2 

azx  +  bsy  +  Csz  as  63  C3 

a^x  +  biy  +  c^z  a^  64  C4 


=  0. 


4.   Reduce  the  following  determinant  to 

one  in  which  all  the  elements 

of  the  first  column  are 

1: 

2     4 

1    3 

3    7 

5    6 

2    0 

0    5 

6     1 

2     3 

5.   Show  that 

(6  +  c)2 

a2 

«2 

(a) 

62 

(c  +  a)2 

62 

=  2  abc(a  +  6  +  c)^  ; 

c2 

c2         (a  +  6)2 

66'  +  cc' 

ba' 

ca' 

(&) 

ab' 

cc'  +  aa' 

c6' 

=  4  aa'bb'cc'. 

ac' 

be' 

aa 

'+66' 

(Hint.     Multiply  the  rows  by  a,  6,  c,  respectively.) 


80 


PLANE  ANALYTIC  GEOMETRY 


[V,§71 


71.  Minors  and  Cofactors.  if  in  a  determinant  both  the  row 
and  column  in  which  any  particular  element  atu  occurs  be  struck  out,  the 
remaining  elements  form  a  determinant  of  order  n  —  1,  which  is  called 
the  minor  of  the  element  aiu- 

From  the  definition  (§  69),  we  observe  that  the  expansion  of  any  de- 
terminant is  linear  and  homogeneous  in  the  elements  of  any  one  row 
(column).  The  terms  which  contain  an  as  factor  are  those  terms  whose 
other  elements  have  all  possible  permutations  of  either  the  first  or  second 
subscripts  2,  3,  •••  n.  Hence  the  sum  of  the  terms  that  contain  an  as 
factor  can  be  expressed  as  an  multiplied  by  its  minor,  i.e. 

«22  •••    din 

«n      •     •     • 

dfii  "'   (Inn 

By  interchanging  the  first  and  second  rows  ((4)  §  70)  we  observe  similarly 
that  the  sum  of  those  terms  which  contain  a-n  as  factor  can  be  written 

«i2  •••  ax, 
—  a'ix     •     •     • 

(ln2  '"  Oni. 

Those  terms  which  contain  a^i  as  factor  are  given  by 

«12  ••'  «i« 
asi      .     •     . 

a„2  •••  a, 

and  so  on.     Hence  the  expansion  of  a  determinant  by  minors  of  the  first 
column  is 


«ii 


a22  •••  a2r. 


«n2 


—  «21 


«12  •••  «ln 


«n2 


+  ...  (-  l)"+^a«i 


«12 


au 


(hi-li  2  ""O^n-lj  n 

Let  Aik  denote  the  cofactor  of  atk  ;  that  is,  (—  1)»+*  times  the  minor  of  Qik] 
and  A  the  original  determinant ;  we  can  then  write  this  expansion  in  the 
form 

A  —  a\\A\\  4-  a>2iA2i  +  ^si^si  +  •••  +  «ni^ni. 

Similarly  by  cofactors  of  the  elements  of  any  column, 

A  =  aikAik  +  a2*^2*  +  «3*^3fe  +  •••  +  ankA„k,  for  A:  =  1,  2,  3,  ...  n, 
and  by  cofactors  of  the  elements  of  any  row, 

A  =  anA  i  +  a<2^i2  +  aoAis  -\-  ...  +  ainAtn,  for  i  =  1,  2,  3,  ...  n. 


V,  §  74]         DETERMINANTS  OF  ANY  ORDER  81 

The  evaluation  of  a  determinant  of  order  n  is  tlius  reduced  to  the 
evaluation  of  n  determinants  of  order  n  —  I.  To  each  of  these  the  same 
process  is  applied  until  determinants  of  order  3  are  obtained  which  can  be 
evaluated  by  the  rule  of  §  42. 

72.  In  case  of  numerical  determinants  this  process  of  successive  reduc- 
tion is  very  much  simplified  by  reducing  to  zero  all  the  elements  of  any 
one  row  (column)  with  the  exception  of  one  element,  say  Uik.  This  can 
always  be  done  by  addition  or  subtraction  of  multiples  of  rows  (columns), 
by  Cor.  2,  §  70.  The  expansion  by  cofactors  of  the  elements  of  this  row 
(column)  then  reduces  to  a  single  term,  viz.  aikAik. 

The  sign  (—  1)«+*  to  be  affixed  to  the  minor  of  au  to  obtain  the  cof actor 
Aik  is  readily  found  by  counting  plus,  minus,  plus,  minus,  etc.,  from  the 
first  element  an  down  to  the  itli  row  and  then  to  the  yfcth  column  until 
Uik  is  reached. 

73.  The  sum  of  the  elements  of  any  row  (column)  multiplied  respec- 
tively by  the  cofactors  of  the  elements  of  any  other  row  (column)  is  zero. 

For,  this  corresponds  to  replacing  the  elements  of  any  row  (column)  by 
the  elements  of  another  row  (column).  Hence  the  determinant  vanishes 
(§  70,  (4),  Cor.  1).  For  example,  if  in  the  expansion  by  cofactors  of  the 
first  row 

aii^n  +  auAu  +  •••  +  otiu^in 
we  replace  the  elements  of  the  first  row  by  those  of  any  other  row  we  find 

anAn  +  ai2^i2  +  •••  +  «m^in  =  0,     for  i  =  2,  3,  •••  w. 

74.  Linear  Equations.  We  write  n  equations  in  n  variables 
aji,  X2,  Xs,  •••  Xn  as  follows, 

auXi  +  ttnXz  +  •••  -f  ainXn  =  h, 

anXl  +  a22X2  +    •••    +  a2nOf7i  =  ^2, 


«nia^l  +  an2X2  +    •••    +  ann^n  =  kn- 

The  determinant  formed  by  the  coefficients  of  the  variables  is  called  the 
determinant  of  the  equations  (§§37,  43)  and  is  denoted  by  A.  To  solve 
the  equations  for  any  one  of  the  variables,  say  Xj,  we  multiply  the  first 
equation  by  the  cof  actor  of  aij  in  A,  i.e.  by  Ay,  the  second  equation  by 
A2j,  the  third  equation  by  Asj,  etc.,  and  add.     This  sum  is  by  §  71 

{aijAij  +  a2jA2j  +  •••  +  anjAnj)Xj  =  Axj  =  kiAij  +  A;2^2j  +  •••  +  knAnj, 
as  the  coefficients  of  all  the  other  variables  vanish  (§  73).      Hence  if 


82 


PLANE  ANALYTIC  GEOMETRY 


[V,  §  74 


^  :5£i  0,  we  have  the  following  rule  :  Each  variable  is  the  quotient  of  two 
determinants,  the  denominator  in  each  case  is  the  determinant  of  the 
equations,  while  the  numerator  is  obtained  from  the  denominator  by  re- 
placing the  coefficients  of  the  variable  by  the  constant  terms  (§§  37,  43). 

75.    Elimination,      if  the  n  linear  equations  are  homogeneous,  i.e.  if 
kiy  ^2,  '"  kn  are  all  zero,  we  have 

«iia-'i  +  aiiX2  +  •••  +  ainXn  =  0, 
a^iXi  +  a22^2  +  •••  4-  «2naJn  =  0, 


an\Xi  +  a„2aJ2  +  •••  +  ann^n  =  0. 
These  equations  are  evidently  satisfied  by  the  values 

Xi  =  0,    iC2  =  0,  •  •  •  iCw  =  0. 

Other  values  of  the  variables  can  satisfy  the  equations  only  if  the  deter- 
minant of  the  equations  is  zero.  For,  the  method  of  §  74  gives  in  the 
case  of  homogeneous  equations 

Axi  =  0,  Axi  =  0,  •••  Axn  =  0. 

Hence  if  Xi,  X2,  •••  Xn  are  not  all  zero  we  must  have 

^  =  0. 
This  result  may  also  be  stated  as  follows :    The  result  of  eliminating  n 
variables  between  n  homogeneous  linear  equations  is  the  determinant  of 
the  equations  equated  to  zero. 

If,  for  instance,  Xn  =^  0,  we  can  divide  each  equation  by  Xn  and  then 
solve  any  n—1  equations  for  the  quotients  Xi/Xn-,  Xz/Xn-,  •••  Xn-i/Xn.  It 
thus  appears  as  in  §  48  that  when  ^  =  0  the  ratios  of  the  variables  can 
be  found  unless  all  the  cofactors  Aij  are  zero. 


EXERCISES 


1.   Show  that 

«ii 

«12      «13      ai4 

an    an  _ 

a2i 

^22      «23      «24 

an    «22 

0 

0       1      au 

0 

0       0       1 

2.    Write  the  expansion  of 

X        0 

0    as 

-1            X 

0      02 

0    -1 

X    a\ 

0 

0 

—  1    ao 

V,  §  76]        DETERMINANTS  OF  ANY  ORDER 


83 


3.  Express  aox*  +  cli^^  +  0,2^'^  +  «3aJ 

4.  Find  the  value  of 


054  as  a  determinant. 


a 

b 

c    d 

—  a 

b 

X    y 

—  a 

-b 

c    z 

—  a 

-b 

-c    d 

5.   Show  that 

1  +  a        1           1 

1 

1 

1        1  +  &        1 

1 

1 

1            1        1  + 

c        1 

1 

=  abcde 

1            11 

1  +  ^ 

1 

111 

1        1  +  e 

6.    Solve  the  equations  : 

Sx+y- z 

-2w=-S 

, 

(a)   . 

2x-y+5 
5x  +  4y- 

z-Sw=6 

z  +  w  =  7, 

> 

(&) 

.    x+2y- 

Sz  + 

w  —- 

3. 

abcde  (1+^  +  1  +  ^ +  1  +  1) 
\       a  -  b     c     d     e  J 


ix-2y  +  2z  +  w  =  ly 
2x  +  Sy-Sz  +  Sw  =  2, 
X  —  y+z  —  4:W=^y 
Sx  +  y-4:z  +  Sw=-5. 

7.  Are  the  following  equations  satisfied  by  other  values  of  the  variables 
than  0,  0,  0,  0  ? 


(a) 


Sx-4:y  +  5z+w  =  0, 
5x  +  2y  —  Sz-io  =  0, 
X  —  y  +  z  +  w  =  0, 
2x  +  2y-3z  +  Sw  =  0. 


(&) 


[Sx  +  2y  +  z-6w  =  0, 
9x  +  9y  +  6z-l0w=0, 
2x  +  y  -  z  +  Sw  =  0, 
x  +  2y  +  z  +  iw  =  0. 


8.  The  relations  between  the  sides  and  cosines  of  the  angles  of  a  tri- 
angle are  a  =  6  cos  7  4-  c  cos  /3,  &  =  c  cos  a  +  a  cos  7,  c  =  a  cos  /3  +  6  cos  a  ; 
find  the  relation  between  the  cosines  of  the  angles. 

76.    Special  Forms,     in  any  determinant 
an  •••  ain 


two  elements  are  called  cowjw^aie  when  one  occupies  the  same  row  and 
column  that  the  other  does  column  and  row  respectively  ;  thus  the 
element  conjugate  to  anc  is  aui.  The  elements  with  equal  subscripts  an, 
a22,  •••  ann  are  called  the  leading  elements;  they  are  their  own  conju- 


84 


PLANE  ANALYTIC  GEOMETRY 


[V,  §  76 


A  determinant  in  which  each  element  is  equal  to  its  conjugate  (i.e. 
ttik  =  aki)  is  called  symmetric. 

A  determinant  in  which  each  element  is  equal  and  opposite  in  sign  to 
its  conjugate  (i.e.  aik  =  — au)  is  called  skew-symmetric \  the  condition 
implies  that  the  leading  elements  are  all  zero. 

A  skeio-symmetric  determinant  of  odd  order  is  always  equal  to  zero. 

For,  if  we  change  the  rows  to  columns  (§70,  Prop.  3)  and  multiply 
each  column  by  —  1,  the  determinant  resumes  its  original  form.  But 
since  the  determinant  is  of  odd  order  we  have  multiplied  by  —  1  an  odd 
number  of  times,  which  changes  the  sign  of  the  determinant  [(4),  §  70]. 
Hence  denoting  the  value  of  the  determinant  by  ^4,  we  have  —  A  —  A, 
i.e.  A=0. 


77.    Multiplication,      it  can  easily  be  verified  for  determinants  of 
the  second  order  that  the  product  of  any  two  such  determinants 

«ii    .«i2|      f>n     hi 

(221       ^22  I         ^21       ^22 

can  be  expressed  as  a  determinant  of  the  second  order  in  any  one  of  the 
four  following  forms  : 


«ii?>ii  +  cinbu 
(i2il>n  +  a22&i2 

^ii^ii  -f  a2i&2i 
ai2?>ii  +  a22&2i 


dnbii  +ai2&22 

«21&21  +  a22'^22 

ail&12  +  «21&22 
ai2&12  4-  ^22^22 


anbii  4-  aiibii    dnbn  +  012622 

a21?>ll  +  dllbil      a2lbi2  +  a22&22 


ail^U  +  «21&12 

dnbn  +  diibii 


anbii  +  021^22 1 

^12^21  +  a22&22 


Thus  the  first  of  these  forms  is,  by  (6),  §  70,  equal  to  the  sum  of  four 
determinants 


«ii&ii 
021611 


011621!      |aii6ii 
021621!      I021611 


012622 
^22622 


012612 
022612 


011621 
O21621 


012612 
O22612 


012622 ] 
022622 


of  which  the  first  and  fourth  are  zero,  while  the  sum  of  the  second  and 
third  reduces  to 


611622 


For  determinants  of  higher  order  the  same  method  can  be  shown  to 
hold.  Without  giving  the  general  proof  we  here  confine  ourselves  to 
illustrating  the'  metho'd  for  determinants  of  the  third  ord^r : 


On     012 
a2i     O22 

—  612621 

On     012 
021     022 

= 

«ii 
021 

012 
O22 

611 
621 

612 
622 

V,  §  77]        DETERMUSTANTS  OF  ANY  ORDER 


85 


an    «i2    «i3 

ftll       &12      &13 

Cii      C12      Ci3 

^21      «22      «23 

&21       &22       &23 

= 

C21      C22      C23 

«31      «32      «33 

631       &32      &33 

C31      C32      C33 

where 

Cn  =  «11&11  +  «12&12  +  «13&13»     C12  =  ail&21  +  «12&22  +  «13&23» 
Cl3  =  ail&31  +  Clnhi  +  «13&33,     C21  =  a2lbn  +  «22&12  +  «23^13i 

etc.  The  product  determinant  can  here  also  be  written  in  four  different 
forms,  according  as  we  combine  rows  with  rows,  rows  with  columns, 
columns  with  rows,  or  columns  with  columns. 

If  the  two  determinants  to  be  multiplied  are  not  of  the  same  order, 
they  can  be  made  of  the  same  order  by  adding  to  the  lower  determinant 
columns  and  rows  consisting  of  zeros  and  a  one  ;  thus 


1    0     0 

a    h 

0    a    & 

z=. 

c    d^ 

0    c     d 

etc. 


EXERCISES 

1.  Show  that  (a)  The  minors  of  the  leading  elements  of  a  symmetric 
determinant  are  symmetric.  (6)  The  minors  of  the  leading  elements  of 
a  skew-symmetric  determinant  are  skew-symmetric,  (c)  The  square  of 
any  determinant  is  a  symmetric  determinant. 

2.  Expand  the  symmetric  determinants : 


(«) 


('0 


H  G 
B  F 
F     C 


(&) 


0 

1 

1 

1 

0 

X 

1 

X 

0 

1 

y 

z 

x  4-  p    px  +  q 


x+p 

px  -^q 

0 


Show  that 


(a) 


0 

1 

1 

1 

a 

b 

= 

1 

c 

d 

0  1  11 

1  a-{-  a    &  +  /3 
1     c+  a    d  +  p\ 


(e) 


(0 


1     X 
X     1 

y    0 

z    0 

1 
1 
1 
0 

1 


y   z 
0    0 


(State  this  property  in  words.) 


86 


PLANE  ANALYTIC  GEOMETRY 


[V,  §  77 


(?>) 


X    a 

a    a 

X    a    a 

a    X    a 

=  {x-ay\x-v2a). 

(c) 

a    X 
a    a 

a    a 
X    a 

a    a    X 

a    a 

a    X 

=  (a;-ffl)3(a;-f-3a). 


4.  Show  that  any  determinant  whose  elements  on  either  side  of  the 
principal  diagonal  are  all  zero,  is  equal  to  the  product  of  the  leading 
elements. 

5.  A  symmetric  determinant  in  which  all  the  elements  of  the  first 
row  and  first  column  are  1  and  such  that  every  other  element  is  the  sum 
of  the  element  above  and  the  element  to  the  right  of  it,  has  the  value  1. 
Illustrate  this  proposition  for  a  determinant  of  the  fourth  order. 

6.  Show  that  any  skew-symmetric  determinant  of  order  2  or  4  is  a 
perfect  square.  This  is  true  for  any  skew-symmetric  determinant  of 
even  order, 

7.  Expand  the  following  determinants : 
0        a        be 

-a  0  f  e 
-h  -f  (id 
— c     —e     —d     0 


(«) 


1        a 

-a        1 

h    -c 


(&) 


•  (c) 


0 

a 

-6 

—  a 

0 

/ 

h 

-/ 

0 

—  c 

—  e 

-d 

8.   Express  as  a  determinant 
0 


(a) 


id) 


d\ 


(&) 


(c) 


X    a    a 

0    0    1 

a    X    a 

. 

1    0    0 

a    a    X 

0    1    0 

(0 


X 

0 

z 

X 

y 

0 

0 

y 

z 

-11 
si 


(/) 


an  —  S      ai2  G5l3 

«31  «32  Cf33 


9.    Show  that 


a    b     c 

d    e    f 

• 

g    h    k 

d' 


b'     c' 

e'     f 

= 

h'    k' 

aii4 

s 

«12 

an 

^21 

«22  +  S 

«23 

an 

«32 

a33  4- 

a    b 

c 

0 

0 

0 

d    e 

f 

0 

0 

0 

g    h 

k 

0 

0 

0 

a    p 

7 

a' 

b' 

c' 

8      e 

f 

d' 

e' 

f 

V    e 

K 

g' 

h' 

k' 

CHAPTER   VI 


y 

( 

h 

I            1    X 

—0 

Fig.  31 


THE   CIRCLE.     QUADRATIC  EQUATIONS 

78.   Circles.     A  circle,  in  a  given  plane,  is  defined  as  the  locus 
of  all  those  points  of  the  plane  which  are 
at  the  same  distance  from  a  fixed  point. 

Let  C  (h,  k)  be  the  center,  r  the  radius 
(Fig.  31) ;  the  necessary  and  sufficient 
condition  that  any  point  P  (x,  y)  is  at 
the  distance  r  from  C  (h,  k)  is  that 

(1)  (a?  -  hfj^{y  _  Jc)^=r^, 

This  equation,  which  is  satisfied  by  the  coordinates  x,  y  of 
every  point  on  the  circle,  and  by  the  coordinates  of  no  other 
point,  is  called  the  equation  of  the  circle  of  center  C  (h,  k)  and 
radius  r. 

If  the  center  of  the  circle  is  at  the  origin  0  (0,  0),  the  equation 
of  the  circle  is  evidently 

(2)  ar^+y  =  r2. 


EXERCISES 
Write  down  the  equations  of  the  following  circles  : 

(a)   center  (3,  2),  radius  7  ; 
(6)    center  at  origin,  radius  3  ; 

(c)  center  at  (—  a,  0),  radius  a  ; 

(d)  circle  of  any  radius  touching  the  axis  Ox  at  the  origin  ; 

(e)  circle  of  any  radius  touching  the  axis  Oy  at  the  origin. 
Illustrate  each  case  by  a  sketch. 

87 


88  PLANE  ANALYTIC  GEOMETRY  [VI,  §  79 

79.  Equation  of  Second  Degree.  Expanding  the  equation 
(1)  of  §  78,  we  obtain  the  equation  of  the  circle  in  the  new  form 

x^  -\-  y""  ~2hx-2ky  -^-h}  +  l?  -  r^  =  0. 
This  is  an  equation  of  the  second  degree  in  x  and  y.     But  it  is  of 
a  particular  form.     The  general  equation  of  the  second  degree 
in  X  and  y  is  of  the  form 

(3)  Ax'^^  Ilxy  +  By^  +  2Ox-\-2Fy-\-C=^0; 

i.e.  it  contains  a  constant  term,  (7;  two  terms  of  the  first  de- 
gree, one  in  x  and  one  in  y ;  and  three  terms  of  the  second  de- 
gree,  one  in  a^,  one  in  xy,  and  one  in  y\ 
If  in  this  general  equation  we  have 

it  reduces,  upon  division  by  A,  to  the  form 

^  +  f+^x  +  ^^y  +  ^  =  0, 

which  agrees  with  the  form  (1)  of  the  equation  of  a  circle,  ex- 
cept for  the  notation  for  the  coefficients. 

We  can  therefore  say  that  any  equation  of  the  second  degree 
which  contains  no  xy-term  and  in  which  the  coefficients  of  a?  and 
y^  are  equal,  may  represent  a  circle. 

80.  Determination  of  Center  and  Radius.  To  draw  the 
circle  represented  by  the  general  equation 

(4)  Ax^  +  Ay^  ^2Gx-\-2Fy^C  =  0, 

where  A,  G,  F,  C  are  any  real  numbers  while  ^  ^^  0,  we  first 
divide  by  A  and  complete  the  squares  in  x  and  y ;  i.e.  we  first 
write  the  equation  in  the  form 

,  GW  f    ,  FY     G\  F""      0 

'^'-aJ^'^aJ-a^^a^'a 

The  left-hand  member  represents  the  square  of  the  distance  of 
the  point  (x,  y)  from  the  point  {—G/A,  —F/A)\  the  right- 


VI,  §  81]     THE  CIRCLE.     QUADRATIC  EQUATIONS       89 

hand  member  is  constant.    The  given  equation  therefore  repre- 
sents the  circle  whose  center  has  the  coordinates 


h-  ^  k-  ^ 


and  whose  radius  is 


This  radius  is,  however,  imaginary  \i  G^  -{-  F"^  <,  AG ]  in  this 
case  the  equation  is  not  satisfied  by  any  points  with  real  co- 
ordinates. 

If  G^  +  F^  =  AG,  the  radius  is  zero,  and  the  equation  is  satis- 
fied only  by  the  coordinates  of  the  point  ( —  G/A,  —  F/A). 

If  G^+F^  >  AG,  the  radius  is  real,  and  the  equation  repre- 
sents a  real  circle. 

Thus,  the  general  equation  of  the  second  degree  (3),  §  79,  repre- 
sents a  circle  if,  and  only  if, 

A  =  B^O,'H^O,  G'  +  F'>AG. 

81.  Circle  determined  by  Three  Conditions.  The  equation 
(1)  of  the  circle  contains  three  constants  h,  k,  r.  The  general 
equation  (4)  contains  four  constants  of  which,  however,  only 
three  are  essential  since  we  can  always  divide  through  by  one  of 
these  constants.  Thus,  dividing  by  A  and  putting  2  G  jA  =  a, 
2  F/A  =  b,  C/A  —  c,  the  general  equation  (4)  assumes  the  form 

(5)  ay'^.f^axi-by-^c^O, 

with  the  three  constants  a,  b,  c. 

The  existence  of  three  constants  in  the  equation  corresponds 
to  the  possibility  of  determining  a  circle  geometrically,  in  a 
variety  of  ways,  by  three  conditions.  It  should  be  remembered 
in  this  connection  that  the  equation  of  a  straight  line  contains 
two  essential  constants,  the  line  being  determined  by  two 
geometrical  conditions  (§  30). 


90  PLANE  ANALYTIC  GEOMETRY  [VI,  §  81 

EXERCISES 

1.  Draw  the  circles  represented  by  the  following  equations: 

(a)  2x^  +  2y^-Sx  +  5y  +  l=0.     (b)  Sx^-\- Sy^+IT  x  -  16y-6  =  0. 
(c)  4 ic2  +  4 2/2 _ 6 X  -  10  y  +  4  =  0.     (d)  x^  +  y^  -\- x  -  4:y  =0. 
(e)  2 x2  +  2  2/2 -  7 a;  =  0.  (f)x^-{-y^-Sx-6z=0. 

2.  What  is  the  equation  of  the  circle  of  center  {h,  k)  that  touches  the 
axis  Ox  ?  that  touches  the  axis  Oy  ?  that  passes  through  the  origin  ? 

3.  What  is  the  equation  of  any  circle  whose  center  lies  on  the  axis 
Ox  ?  on  the  axis  Oy?  on  the  line  y=  x?  on  the  line  y  =  2x?  on  the  line 
y  =  mx  ? 

4.  Find  the  equation  of  the  circle  whose  center  is  at  the  point  (—  4,  6) 
and  which  passes  through  the  point  (2,  0). 

5.  Find  the  circle  that  has  the  points  (4,  —  3)  and  (  —  2,  —  1)  as  ends 
of  a  diameter. 

6.  A  swing  moving  in  the  vertical  plane  of  the  observer  is  48  ft.  away 
and  is  suspended  from  a  pole  27  ft.  high.  If  the  seat  when  at  rest  is  2  ft. 
above  the  ground,  what  is  the  equation  of  the  path  (for  the  observer  as 
origin)?  What  is  the  distance  of  the  seat  from  the  observer  when  the 
rope  is  inclined  at  45^  to  the  vertical  ? 

7.  Find  the  locus  of  a  point  whose  distance  from  the  point  (a,  h)  is  /c 
times  its  distance  from  the  origin. 

Let  P  (ic,  y)  be  any  point  of  the  locus  ;  then  the  condition  is 


V(x-a)2-f  (2/-6)2=  K  Vx2  +  2/2  ; 

upon  squaring  and  rean*anging  this  becomes  : 

(1  -  k2)x2  +  (1  -  k2)2/2 -2ax-2hy  -\-  cfi-\-  62  =  o. 

Hence  for  any  value  of  k  except  k  =  1,  the  locus  is  a  circle  whose  center  is 
a/{\  -  k2),  6/(1  -  k2)  and  whose  radius  is  k  y/d^  +  6V(1  -  k^).  What 
is  the  locus  when  ic  =  1  ? 

8.  Find  the  locus  of  a  point  twice  as  far  from  the  origin  as  from  the 
point(6,  —  3).     Sketch. 

9.  What  is  the  locus  of  a  point  whose  distances  from  two  points  Pi, 
P2  are  in  the  constant  ratio  k  ? 


VI,  §82]     THE  CIRCLE.    QUADRATIC  EQUATIONS       91 

10.  Determine  the  locus  of  the  points  which  are  k  times  as  far  from 
the  point  (—2,  0)  as  from  the  point  (2,  0).  Assign  to  k  the  values 
\/5,  V8,  V2,  I VS,  ^  \/3,  I  \/2  and  illustrate  with  sketches  drawn  with 
respect  to  the  same  axes. 

11.  Determine  the  locus  of  a  point  whose  distance  from  the  line 
Sx  — 4y+l=0  is  equal  to  the  square  of  its  distance  from  the  origin. 
Illustrate  with  a  sketch. 

12.  Determine  the  locus  of  a  point  if  the  square  of  its  distance  from 
the  line  x  +  y  —  a  =  0  is  equal  to  the  product  of  its  distances  from  the 
axes. 

82.  Circle  in  Polar  Coordinates.  Let  us  now  express  the 
equation  of  a  circle  in  polar  coordinates.  If  (7(ri,  <^i)  is  the 
center  of  a  circle  of  radius  a  (Fig.  32) 
and  P(rj  <^)  any  point  of  the  circle, 
then  by  the  cosine  law  of  trigo- 
nometry 0^"'-^  T'^. 

r^  +  ri^  —  2  riV  cos  (<^  —  <^i)  =  a\  Fig.  32 

This  is  the  equation  of  the  circle  since,  for  given  values  of  ?-i, 
<^i,  a,  it  is  satisfied  by  the  coordinates  r,  4>  of  every  point  of 
the  circle,  and  by  the  coordinates  of  no  other  point. 
Two  special  cases  are  important: 

(1)  If  the  origin^ be  taken  on  the  circumference  and  the 
.polar  axis  along  a  diameter  OA  (Fig.  33), 
the  equation  becomes 

^2  _f_  a2  —  2  ar  cos  <f>  =  a^, 
i.e.  r  =  2  a  cos  <^. 

This  equation  has  a  simple  geometrical 

interpretation :  the  radius  vector  of  any 

point  Pon  the  circle  is  the  projection  of  the  diameter  OA  =2  a 

on  the  direction  of  the  radius  vector. 

(2)  If  the  origin  be  taken  at  the  center  of  the  circle,  the 
equation  is  r  =  a. 


% 


92  PLANE  ANALYTIC  GEOMETRY         [VI,  §  82 

^  EXERCISES 

1.  Draw  the  following  circles  in  polar  coordinates  : 

'"^(a)  r  =  10  cos  0.    '~~-  (b)  r  =  2a  cos  (0  —  ^  ir).  (c)  r  —  sin  0. 

((?)  r  =  6.  (e)  r  =  7  sin  (0  —  |  tt)  .  {f)r-  17  cos  0. 

2.  Write  the  equation  of  the  circle  in  polar  coordinates  : 
\a)  with  center  at  (10,  ^tt)  and  radius  5  ; 

(6)  with  center  at  (6,  \  tt)  and  touching  the  polar  axis  ; 

(c)  with  center  at  (4,  |  tt)  and  passing  through  the  origin  ; 

(d)  with  center  at  (3,  tt)  and  passing  through  the  point  (4,  \  tt)  . 

■^      3.   Change  the  equations  of  Ex,  (1)  and  (2)  to  rectangular  coordinates 
with  the  origin  at  the  pole  and  the  axis  Ox  coincident  with  the  polar  axis. 
4.   Determine  in  polar  coordinates  the  locus  of  the  midpoints  of  the 
chords  drawn  from  a  fixed  point  of  a  circle. 

83.  Quadratic  Equations.  The  fundamental  problem  of 
finding  tlie  intersections  of  a  line  and  a  circle  leads,  as  we  shall 
see  (§  86),  to  a  quadratic  equation.  Before  discussing  it  we 
here  recall  briefly  the  essential  facts  about  quadratic  equations. 

The  method  for  solving  a  quadratic  equation  consists  in  com- 
pleting the  square  of  the  terms  in  x^  and  a*,  which  is  done  most 
conveniently  after  dividing  the  equation  by  the  coefficient  of  x^. 

The  equation 

«2  +  2  j9a;  +  g  =  0 
has  the  roots 

x=.  —  J)  ±  V/52  —  q. 

The  quantity  i[P-  —  q\.^  called  the  discriminant  of  the  equation. 
According  as  the  discriminant  is  positive,  zero,  or  negative,  the 
roots  are  real  and  different,  real  and  equal,  or  imaginary.  In 
the  last  case,  i.e.  when  p^  <  g,  the  roots  are,  more  precisely, 
conjugate  complex,  i.e.  of  the  form  a  +  bi  and  a  —  hi,  where  a 
and  h  are  real  while  i  =  V—  1. 

As  remarked  above,  any  quadratic  equation  may  be  thrown 
into  the  form  here  discussed,  by  dividing  by  the  coefficient 
of  x^. 


VI,  §  84]     THE  CIRCLE.     QUADRATIC  EQUATIONS       93 

84.  Relations  between  Roots  and  Coefficients.    If  we  de- 
note the  roots  of  the  quadratic  equation 

by  Xi  and  x^ ,  we  have 

Xi=  —  p  -\-  Vp^  —  q,        X2  =  —  p  —  Vi?2  —  q, 
whence 

Xi -{- x^  =  —  2  p,     X1X2  =  q ; 

i.  e.  the  sum  of  the  roots  of  a  quadratic  equation  in  which  the 

coefficient  ofx^  is  reduced  to  1  is  equal  to  minus  the  coefficient  of 

x;  the  product  of  the  roots  is  equal  to  the  constant  term. 

With  the  values  of  x^,  x.2  just  given  we  find 

{x  —  x^{;x  —  x^  =  0?  -\-  2px  +  g, 

so  that  the  quadratic  equation  can  be  written  in  the  form 

{X  —  X^{X  —  CCg)  =  0, 

which  gives 

These  properties  of  the  roots  often  make  it  possible  to  solve 
a  quadratic  equation  by  inspection. 

EXERCISES 
1.   Solve  the  quadratic  equations  : 

(a)  a:2  -  6  X  +  8  =  0.  I  (ft)  x2  +  5  a;  -  14  =  0. 

(c)  2  a;2  -  x  -  28  =  0.        '  (d)  6  jc^  -  7  a;  -  6  =  0. 

(e)  a;2  +  2  &x  -  a^  +  &2  ^^  q.  {f)  a^x^  -  {a^  +  b^)x  +  b'^  =  0. 

(fir)  ax2  +  &x  =  0.  (h)  12  a:2  +  8  x  -  15  =  0. 

/  2.    Show  that  the  solutions  of  the  quadratic  equation  ax'^  +  &x  +  c  =  0 
may  be  written  in  the  form  x  =  -  -^  ±  ^^  -  4  «c 


2a  2a 

When  are  these  solutions  real  and  unequal  ?  equal  ?  imaginary  ? 

3.    Write  down  the  quadratic  equation  that  has  the  following  roots : 
(a)  3,  -  2.  (ft)   -  3,  0.  (c)  5,  -  5. 

(d)  a-b,  a  +  b.         (e)  3  -  2V3,  3  +  2  >/3.        (/)  1  +  \/2,  1  -  \/2^ 

(9)  c,  -i.  (h)  h-h  '        (i)  3,  V2. 


94  PLANE  ANALYTIC  GEOMETRY  [VI,  §  84 

4.    Without  solving,  determine  the    nature  of  the  roots  of  the  follow- 
ing equations  : 

»  5a;2-6x-2  =  0.  (6)     9x^  +  (>x+lz=0. 

^c)  2  a:2  -  a;  +  3  =  0.  (cZ)     20  a;2  +  6  a;  -  5  =  0. 

(e)  llx2-4x-^^  =  0.  (/)     3a:2  +  2x  +  l  =  0. 

6-   For  what  values  of  k  are  the  roots  of  the  following  equations  real 
and  different  ?  real  and  equal  ?  conjugate  complex  ? 

(a)  x2-  4x  +  A:  =  0.  (6)  a:2  +  2  ^•a;  +  36  =  0. 

i^  ,(c)  9x^  +  kx+26  =  0.  (d)  ax^-\-bx  +  k  =  0. 

"^  (e)  A:x2  -  5  X  +  6  =  0.  (/)  ax2  +  A:x  +  c  =  0. 

6.  Solve  the  following  equations  as  quadratic  equations  : 
(_(a)  ?/4_3y2_4^0.    (Let  1/2=  2;.)        (6)  2;3-2  +  3  2;-i  -  2  =  0. 

, ,  ,x       2       ,  X  +  3     „ 

(c)  X  +  V^TTS  =  3.  (^)  ^^  +  -^  =  2. 

(e)  m6  +  18  m3  -  243=  0.  (/)  2  x"!  +  x'^  -  16  =  0. 

7.  If    xi  and  X2  are  the  roots  of  x2  +  2  px  +  g  =  0,  find  the  values  of 

(7(a)    Xi2x2  +  X1X22.  (6)    Xi2  +  X22.  (C)    (Xi  -  X2)2. 

Xi        X2  Xi2        X22 

and  apply  these  results  to  the  case  x2  —  3  x  +  4  =  0. 

8.  Without  solving,  form  the  equation  whose  roots  are  each  twice 
the  roots  of  x2  -  3  x  +  7  =  0.     [See  §  84.] 

9.  What  is  the  equation  whose    roots    are  m  times  the  roots    of 
x2  +  2px  +  ^  =  0? 

10.   Form  the  equation  whose  roots  are  related  to  the  roots  of  2  x2  — 
3  X  —  5  =  0,  in  the  following  ways  : 

(o)  less  by  2  ;  (h)  greater  by  3  ;  (c)  divided  by  6. 

85.  Simultaneous    Linear  and  Quadratic  Equations.    To 

solve  two  equations  in  x  and  y  of  which  one  is  of  the  first 
degree  (linear)  while  the  other  is  of  the  second  degree,  it  is 
generally  most  convenient  to  solve  the  linear  equation  for  either 
X  or  y  and  to  substitute  the  value  so  found  in  the  equation  of  the 
second  degree.  It  then  remains  to  solve  a  quadratic  equation. 
An  equation  of  the  first  degree  represents  a  straight  line. 


VI,  §86]     THE  CIRCLE.    QUADRATIC  EQUATIONS       95 

If  the  given  equation  of  the  second  degree  be  of  the  form 
described  in  §  79,  it  will  represent  a  circle.  By  solving  two 
such  simultaneous  equations  we  find  the  coordinates  of  the 
points  that  lie  both  on  the  line  and  on  the  circle,  i.e.  the  points 
of  intersection  of  line  and  circle. 

86.   Intersection  of  Line  and  Circle.    Let  us  find  the  in- 
tersections of  the  line 

y  =  mx  -h  b 

with  the  circle  about  the  origin 

Substituting  the  value  of  y  from  the  former  equation  into  the 
latter,  we  find  the  quadratic  equation  in  x : 

x^+(mx-{-by=r^, 
or  (1  +  'nv')^  +  2  mbx  -\-b^-r^=:0' 

The  two  roots  Xi,  X2  of  this  equation  are  the  abscissas  of  the 
points  of  intersection ;  the  corresponding  ordinates  are  found 
by  substituting  iCi,  X2in  y  =  mx  +  b. 

It  is  easily  seen  that  the  abscissas  Xi,  x^  are  real  and  differ- 
ent if  (l  +  mV-62>o, 

.      .0  b         ^ 

I.e.  II  — ^:=z=:  <  r. 

Vl  +  rn? 


Since  m  =  tan  a,  and  hence  1/ Vl  +  m^  =  cos  a,  the  preceding 
relation  means  that  b  cos  a  <  r,  i.e.  the  line  has  a  distance  from 
the  origin  less  than  the  radius  of  the  circle.     If 

the  roots  x^,  x^  are  real  and  equal.  The  line  and  the  circle  then 
have  only  a  single  point  in  common.  Such  a  line  is  said  to 
touch  the  circle  or  to  be  a  tangent  to  the  circle.     If 

(1  +  'rri')7^  -b^<0, 
the  roots  are  complex,  and  the  line  has  no  points  in  common 
with  the  circle. 


96  PLANE  ANALYTIC  GEOMETRY  [VI,  §  87 

87.  The  General  Case.  The  intersections  of  the  line  and 
circle 

i»^  +  2/^  +  «i»  +  &2/  +  c  =  0, 

are  found  in  the  same  way :  substitute  the  value  of  y  (or  a;), 
found  from  the  equation  of  the  line,  in  the  equation  of  the 
circle  and  solve  the  resulting  quadratic  equation. 

It  is  often  desired  to  determine  merely  ivhetlier  the  line  is 
tangent  to  the  circle.  To  answer  this  question,  substitute  y 
(or  x)  from  the  linear  equation  in  the  equation  of  the  circle 
and,  without  solving  the  quadratic  equation^  write  down  the  con- 
dition for  equal  roots  (p^  =  q,  §  83). 

EXERCISES 

1.  Find  the  coordinates  of  the  points  where  the  circle  x^  +  y'^^  —  x  -\-  y 
—  12  =  0  crosses  the  axes. 

2.  Find  the  intersections  of  the  line  3aj  +  y— 5  =  0  and  the  ^circle 

x2  +  1/2  _  22  a;  -  4  y  +  25  =  0. 

3.  Find  the  intersections  of  the  line  2x  —  1  y  +  6  =  0  and  the  circle 
2x2  +  2y2  4.9x  +  9?/-ll  =  0. 

4.  Find  the  equations  of  the  tangents  to  the  circle  xr  +  y'^  =  16  that 
are  parallel  to  the  line  y  =—Sx  -j-S. 

5.  Show  that  the  equations  of  the  tangents  to  the  circle  x^  -\-  y"^  =  r^ 
with  slope  m  are  y  =  mx  ±  rVl  +  m'^. 

6.  For  what  value  of  r  will  the  line  3x-2y  —  5  =  0be  tangent  to  the 
circle  x^  +  y^  =  r^  ? 

7.  Find  the  equations  of  the  tangents  to  the  circle  2x'^  +  2y^  —  Sx 
+  5?/  —  7  =  0  that  are  perpendicular  to  the  line  x  +  2y  +  3  =  0. 

8.  Find  the  midpoint  of  the  chord  intercepted  by  the  line  5x-y  +  9=0 
on  the  circle  x^  -\-y^  =  18.     (Use  §  84.) 

9.  Find  the  equations  of  the  tangents  to  the  circle  x^  +  y2  _  53  that 
pass  through  the  point  (10,  4). 


VI,  §89]     THE  CIRCLE.    QUADRATIC  EQUATIONS       97 


88.  The  Tangent  to  a  Circle.  The  tangent  to  a  circle  (com- 
pare §  86)  at  any  point  P  may  be  defined  as  the  perpendicular 
through  P  to  the  radius  passing  through  P.  To  find  the  equa- 
tion of  the  tangent  to  a  circle  whose  center  is  at  the  origin, 

x^  -\-  y"^  =  r^, 
at  the  point  P  (x,  y)  of  the  circle  (Fig.  34),  observe  that  the 
distance  p  of  the  tangent  from  the  origin 
is  equal  to  the   radius  r  and  that  the 
angle  p  made  by  this  distance  with  the 
axis  Ox  is  such  that 

cos  /?  =  - ,  sin  /8  =  -^  : 
T  r 

substituting  these  values  in  the  normal 

form  X  cos  /8  +  r  sin  ^  =  p    of   the  Fig.  m 

equation  of  a  line  (§  54),  we  find  as  equation  of  the  tangent 

xX-]-yY=r'^, 

where  x,  y  are  the  coordinates  of  the  point  of  contact  P  and 
X  Y  are  those  of  any  point  of  the  tangent. 

89.  The  General  Case.  To  find  the  equation  of  the  tangent 
to  a  circle  whose  center  is  not  at  the  origin  let  us  write  the 
general  equation  (4),  §  80,  viz. 

(4)  Ax""  +  .4?/2  +  2  (^a;  +  2  i<V  +  C  =  0, 

in  the  form 

F^      C 


"+fT^i^+2j=^+^^  ^ 


a)     a^ 


where  —  G/A,  —  F/A  are  the  coordinates  of  the  center  and 
Q2/ji  +  F^/A"-  C/A  is  the  square  of  the  radius  r  (§  80). 
With  respect  to  parallel  axes  through  the  center  the  same  circle 
has  the  equation 


2.2  G^    ,    F"" 

-^       A''     A^ 


=  r\ 


98  PLANE  ANALYTIC  GEOMETRY         [VI,  §  89 

and  the  tangent  at  the  point  P{x,  y)  of  the  circle  is  (§  88) : 

Hence,  transferring  back  to  the  original  axes,  we  find  as 
equation  of  the  tangent  at  P  (x,  y)  to  the  circle  (4) : 

AxX-\-AyY-\-G{:x+X)^F{y^-  Y)+  (7=0. 

This  general  form  of  the  tangent  is  readily  remembered  if  we 
observe  that  it  can  be  derived  from  the  equation  (4)  of  the 
circle  by  replacing  x^  by  xX,  y^  hy  yY,  2  a?  by  ic+ X,  2y'byy-\-Y. 

EXERCISES        \  \n 

1.  Find  the  tangent  to  the  given  circle  at  the  given  point : 

(a)  0^2  +  2/2  =  41,  (5,  -4). 

(6)  x^  +  y^  +  Qx  +  ^y- 16  =  0,  (-2,  3). 

(c)  3a-2  +  3?/2  +  10a;  +  17?/+18  =  0,  (-2,  -o). 

(d)  a;2  +  ?/2  -  ax  -hy  =  0,  («,  6). 

2.  The  equation  of  any  circle  through  the  origin  can  be  written  in  the 
form  (§  81)  x^  +  y^  +  ax  +  by  =  0;  show  that  the  line  ax  -\- by  =  0  is  the 
tangent  at  the  origin,  and  find  the  equation  of  the  parallel  tangent. 

3.  Derive  the  equation  of  the  tangent  to  the  circle  {x—h)'^+{y—k)^=:r^. 

4.  Show  that  the  circles  x'^  +  y^  —  6x +  2y  +  2  =  0  and  x^  +  y'^  —  4y 
+  2  =  0  touch  at  the  point  (1,  1). 

5.  Find  the  tangents  to  the  circle  x^  +  y^  —  2x  — 10y-{-9  =  0  at  the 
extremities  of  the  diameter  through  the  point  (—  1,  11/2). 

6.  The  line  2aj  +  2/  =  10  is  tangent  to  the  circle  x^  +  y'^  =  20  ;  what  is 
the  point  of  contact  ? 

7.  What  is  the  point  of  contact  if  Ax -{-  By  -h  C  =  0  is  tangent  to  the 
circle  x^  +  y^  =  r'^? 

8.  Show  that  x  —  y  —  l  =  0  is  tangent  to  the  circle  aj^  +  ?/2  +  4  x 
—  10  ?/  —  3  =  0,  and  find  the  point  of  contact. 

9.  By  §  86,  the  line  y  =  mx  +  &  has  but  one  point  in  common  with 
the  circle  x^  +  ?/2  =  r^  if  ( 1  +  m'^)r^  =  b^  ;  show  that  in  this  case  the  radius 
drawn  to  the  common  point  is  perpendicular  to  the  line  y  =  mx  -\-  b. 


VI,  §  90]     THE  CIRCLE.    QUADRATIC  EQUATIONS       99 


90.  Circle  through  Three  Pomts.  To  fiyid  the  equation  of 
the  circle  passing  through  three  points  Piix^^y^,  -^2(^2?  2/2)5 
A  (^3  J  2/3)?  observe  that  the  coordinates  of  these  points  satisfy 
the  equation  of  the  circle  (§  81) 

(6)  .T2^2/'  +  ^a^  +  %  +  c  =  0; 

hence  we  must  have 


(J) 


^i  +  Vi  +  «^i  +  &2/1  +  c  =  0, 
^2  +  2/2^  +  «^2  +  &2/2  +c  =  0, 
.^i  +  Vz^  +  «%  +  &2/3+  c  =  0. 


From  the  last  three  equations  we  can  find  the  values  of  a,  6, 
and  c ;  these  values  must  then  be  substituted  in  the  first  equa- 
tion. 

In  general  this  is  a  long  and  tedious  operation.  What  we 
actually  wish  to  do  is  to  eliminate  a,  b,  c  between  the  four 
equations  above.  The  theory  of  determinants  furnishes  a  very 
simple  means  of  eliminating  four  quantities  between  four 
homogeneous  linear  equations  (§  75).  Our  equations  are  not 
homogeneous  in  a,  6,  c.  But  if  we  write  the  first  two  terms  in 
each  equation  with  the  factor  1 :  (a?^  -f  y^)  .  1,  (x-^  4-  y-^)  •  1,  etc., 
we  have  four  equations  which  are  linear  and  homogeneous  in  1, 
a,  b,  c ;  hence  the  result  of  eliminating  these  four  quantities  is 
the  determinant  of  their  coefficients  equated  to  zero.  Thus  the 
equation  of  tJie  circle  through  three  points  is 


=  0 


Compare  §  49,  where  the  equation  of  the  straight  line  through 
two  points  is  given  in  determinant  form. 


^  +  y' 

X      y 

1 

aa'  +  2/i' 

Xi     2/1 

1 

3^2^  +  2/2' 

^2       2/2 

1 

x^ty^' 

Xs     2/3 

1 

100  PLANE  ANALYTIC  GEOMETRY  [VI,  §  90 

EXERCISES 

1.   Find  the  equations  of  the  circles  that  pass  through  the  points  : 
^  (a)  (2,3),  (-1,2),  (0,-3). 
-^(6)  (0,0),   (1,-4),  (5,0). 

(c)   (0,  0),   (a,  0),   (0,  b). 

•  2.    Find  the  circles  through  the   points   (3,  —  1),    (—  1,  —2)   which 
touch  the  axis  Ox. 

^    3.   Find  the  circle  through  the  points  (2,  1),  (—  1,  3)  with  center  on 
the  line  3x  —  y  +  2-0. 

4.    Find  the  circle  whose  center  is  (3,  —  2)  and  which  touches  the 
line  3a:  +  4y-12  =  0. 

6.    Find  the  circle  through  the  origin  that  touches  the  line 
4x-5y-  14  =  Oat  (6,  2). 

6.   Find  the  circle  inscribed  in  the  triangle  determined  by   the  lines 

24x-7?/  +  3=0,  3x-4«/-9  =  0,   5x  +  12y-50  =  0. 
7.'  Two  circles  are  said  to  be  orthogonal  if  their  tangents  at  a  point  of 
intersection  are  perpendicular ;  the  square  of  the  distance  between  their 
centers  is  then  equal  to  the  sum  of  the  squares  of  their  radii.     If  the 
equations  of  two  intersecting  circles  are 

x^  -\-y^  +  aix  +  biy  +  Ci  =0,  and  x^  +  y^  +  a^x  +  &22/  +  C2  =  0, 
show  that  the  circles  are  orthogonal  when  aia2  +  6162  =  2(ci  +  C2). 

8.  Find  the  circle  that  has  its  center  at  (—2,  1)  and  is  orthogonal  to 
the  circle  x^  +  y^-6x  + S  =  0. 

9.  Find  the  circle  that  has  its  center  on  the  line  i/  =  3  x  +  4,  passes 
through  the  point  (4,  —  3),  and  is  orthogonal  to  the  circle 

x^  +  y^  +  lSx  +  5y  +  2  =0. 

91.    Inversion.      A  circle  of  center  O  and  radius  a  being  given 
(Fig.  35),  we  can  find  to  every  point  P  of  the  plane 
(excepting  the  center  O)  one  and  only  one  point  P' 
on  OP,  produced  beyond  P  if  necessary,  such  that 

OP .  OP'  =  a2. 

The  point  P'  is  said  to  be  inverse  to  P  with  respect 

to  the  circle  (0,  a)  ;    and  as  the  relation  is  not  Fig.  35 


VI,  §92]     THE  CIRCLE.    QUADRATIC  EQUATIONS     101 


changed  by  interchanging  P  and  P',  the  point  P  is  inverse  to  P'.  The 
point  0  is  called  the  center  of  inversion. 

It  is  clear  that  (1)  the  inverse  of  a  point  P  within  the  circle  is  a  point 
P'  without,  and  vice  versa  ;  (2)  the  inverse  of  a  point  of  the  circle  itself 
coincides  with  it ;  (3)  as  P  approaches  the  center  0,  its  inverse  P'  moves 
off  to  infinity,  and  vice  versa. 

The  inverse  of  any  geometrical  figure  (line,  curve,  area,  etc.)  is  the 
figure  formed  by  the  points  inverse  to  all  the  points  of  the  given  figure. 

92.  Inverse  of  a  Circle.  Taking  rectangular  axes  through  O 
(Fig.  36),  we  find  for  the  relations  between  the  coordinates  of  two  in- 
verse points  P{x,  y),  P'  (x',  y'),  if  we  put  OP  =  r,  OP'  =  r' ; 


X      y      r 

rr' 

a2 
r2 

since  rr'  =  a^  .  hence 

X'-     ^'"^ 

y'-- 

X2+2/2' 

and  similarly 

._     aV 

11 

_    a'^y' 

X'2  +  2/'2 


Fig.  36 


These  equations  enable  us  to  find  to  any  curve  whose  equation  is  given  the 
equation  of  the  inverse  curve,  by  simply  substituting  for  x,  y  their  values. 

Thus  it  can  be  shown  that  hy  inversion  any  circle  is  transformed  into 
a  circle  or  a  straight  line. 

For,  if  in  the  general  equation  of  the  circle 

^(x2  +  y2)  +  2  ^x  +  2  Py  +  (7  =  0 
we  substitute  for  x  and  y  the  above  values,  we  find 


Aa^ 


x'2  +  y' 


+  2(?a2. 


4-2Pa2. 


y' 


4-c  =  o, 


(X'2  +  ?/'2)2    ■  x'-2  +  y'-^  X'2  4-  ?/'2 

that  is,  Aa^  +  2  QaH'  +  2  Fay  +  0(x'2  +  y''^)  =  0, 

which  is  again  the  equation  of  a  circle,  provided  C  ^0.  In  the  special 
case  when  C  =  0,  the  given  circle  passes  through  the  origin,  and  its  in- 
verse is  a  straight  line.  Thus  every  circle  through  the  origin  is  trans- 
formed hy  inversion  into  a  straight  line.  It  is  readily  proved  conversely 
that  every  straight  line  is  transformed  into  a  circle  passing  through  the 
origin  ;  and  in  particular  that  every  line  through  the  origin  is  transformed 
into  itself,  as  is  obvious  otherwise. 


102 


PLANE  ANALYTIC  GEOMETRY 


[VI,  §  92 


EXERCISES 

1.  Find  the  coordinates  of  the  points  inverse  to  (4,  3),  (2,  0),  (—5,  1) 
with  respect  to  the  circle  x^-j-y'^  =  26. 

2.  Show  that  by  inversion  every  line  (except  a  line  through  the  center) 
is  transformed  into  a  circle  passing  through  the  center  of  inversion. 

3.  Show  that  all  circles  with  center  at  the  center  of  inversion  are 
transformed  by  inversion  into  concentric  circles. 

4.  Find  the  equation  of  the  circle  about  the  center  of  inversion  which 
is  transformed  into  itself. 

6.   With  respect  to  the  circle  x^  +  y"^  =  16,  find  the  equations  of  the 
curves  inverse  to  : 

(a)  x=b,     (b)  x-y=0,     (c)  x'^  +  y^-6x=0,      (d)  x^+y^-lOy  +  l=0, 
(e)  Sx-^y-\-Q=0. 

6.  Show  that  the  circle  Ax'^  +  Ay'^  -^2  Gx-\-2  Fy  +  a^A  =  0  is  trans- 
formed into  itself  by  inversion  with  respect  to  the  circle  a:^  +  y2  —  q2^ 

7.  Prove  the  statements  at  the  end  of  §  92, 

93.    Pole  and  Polar.     Let  P,  P'  (Fig.  37)  be  inverse  points  with 
respect  to  the  circle  (O,  a)  ;  then  the  perpen- 
dicular I  to  OP  through  P'  is  called  the  polar  of 
P,  and  P  the  pole  of  the  line  Z,  with  respect  to 
the  circle. 

Notice  that  (1)  if  (as  in  Fig.  37)  P  lies  within 
the  circle,  its  polar  I  lies  outside ;  (2)  if  P  lies 
outside  the  circle,  its  polar  intersects  the  circle 
in  two  points ;  (3)  if  P  lies  on  the  circle,  its 
polar  is  the  tangent  to  the  circle  at  P. 


Fig.  37 


Referring  the  circle  to  rectangular  axes  through  its  center  (Fig.  38)  so 
that  its  equation  is 

x2  -)-  2/2  =  a% 

we  can  find  the  equation  of  the  polar  I  of 
any  given  point  P(ic,  y).  For,  using 
as  equation  of  the  polar  the  normal 
form  X  cos  /3+  F  sin  /3  =;;,  we  have 
evidently,  if  P'  is  the  point  inverse 
toP: 


VI,  §94j     THE  CIRCLE.    QUADRATIC  EQUATIONS     103 


cos/3 


\/x^  +  y'^ 


sin/3 


therefore  the  equation  becomes 
xX 


■vx'^  +  y'^ 


yY    _ 


p=OP'  = 


or  simply 


xX-\-yY=a'^. 


This  then  is  the  equation  of  the  polar  I  of  the  point  P  {x,  y)  with  re- 
spect to  the  circle  of  radius  a  about  the  origin.  If,  in  particular,  the 
point  P  (a;,  y)  lies  on  the  circle,  the  same  equation  represents  the  tan- 
gent to  the  circle  xP-  ■\-'f  —  a^  at  the  point  P  (x^y),  as  shown  previously 
in  §  88. 

94.  Chord  of  Contact.  The  polar  l  of  any  outside  point  P  with 
respect  to  a  given  circle  passes  through  the  points  of  contact  Ci ,  C2  of 
the  tangents  drawn  from  P  to  the  circle. 

To  prove  this  we  have  only  to  show  that  if  Ci  is  one  of  the  points  of 
intersection  of  the  polar  I  of  P  with  the  circle,  then  the  angle  OCiP 
(Fig.  39)  is  a  right  angle.  Now  the  triangles 
OCiP  and  OP'Ci  are  similar  since  they  have 
the  angle  at  0  in  common  and  the  including 
sides  proportional  owing  to  the  relation 

OP  •  OP'  =  a2, 

OP^    a 
a       0P'\ 


i.e. 


where  a  =  OCi.     It  follows  that  ^  OC\P=  j.^^   3^ 

0P'Ci  =  |7r. 

The  rectilinear  segment  C1C2.  is  sometimes  called  the  chord  of  contact 
of  the  point  P.  We  have  therefore  proved  that  the  chord  of  contact  of 
any  outside  point  P  lies  on  the  polar  of  P. 

It  follows  that  the  equations  of  the  tangents  that  can  he  drawn  from 
any  outside  point  P  to  a  given  circle  can  be  found  by  determining  the 
intersections  Ci ,  Ci  of  the  polar  of  P  with  the  circle ;  the  tangents  are 
then  obtained  as  the  lines  joining  Ci ,  C2  to  P. 


104 


PLANE  ANALYTIC  GEOMETRY  [VI,  §  95 


95.  The  General  Case.  The  equation  of  the  polar  of  a  point 
P  (x,  y)  with  respect  to  any  circle  given  in  the  general  form  (4), 
§  80,  viz., 

(4)  Ax^-  +  Ay^-\-2Gx  +  2Fy  +  C  =  0, 

is  found  by  the  same  method  that  was  used  in  §  89  to  generalize  the 
equation  of  the  tangent.  Thus,  with  respect  to  parallel  axes  through  the 
center  the  equation  of  the  circle  is 

C 
A' 
the  equation  of  the  polar  of  P(x,  y)  with  respect  to  these  axes  is  by 


--^-l-f 


§93: 


Hence,  transferring  back  to  the  original  axes,  we  find  as  equation  of  the 
polar  of  P  {x,  y)  with  respect  to  the  circle  (4)  : 

AxX-\-  AyY -{-  G{x  +  X)+  F(y  +  Y)+  C  =  0. 
If,  in  particular,  the  point  P  (x,  y)  lies  outside  the  circle,  this  polar 
contains  the  chord  of  contact  of  P;  if  P  lies  on  the  circle,  the  polar  be- 
comes the  tangent  at  P  (§  89). 

96.  Construction  of  Polars.  if  a  point  Pi  describes  a  line  I,  its 
polar  h  with  respect  to  a  given  circle  (0,  a)  turns  about  a  fixed  point, 
viz.,  the  pole  P  of  the  line  I  (Fig.  40). 
Conversely,  if  a  line  h  turns  about  one 
of  its  points  P,  its  pole  Pi  with  respect 
to  a  given  circle  {0,  a)  describes  a  line  Z, 
viz.  the  polar  of  the  point  P. 

For,  the  line  I  is  transformed  by  in- 
version with  respect  to  the  circle  (0,  a) 
into  a  circle  passing  through  0  and 
through  the  pole  P  of  I;  as  this  circle 
must  obviously  be  symmetric  with  respect 
to  OP  it  must  have  OP  as  diameter.  Any 
point  Pi  of  I  is  transformed  by  inversion 
into  that  point  Q  of  the  circle  of  diameter  OP  at  which  this  circle  is  in- 
tersected by  OPi  .  The  polar  of  Pi  is  the  perpendicular  through  Q  to 
OPi ;  it  passes  therefore  through  P,  wherever  Pi  be  taken  on  ^ 

The  proof  of  the  converse  theorem  is  similar. 


Fig.  40 


VI,  §96]     THE  CIRCLE.    QUADRATIC  EQUATIONS     105 

The  pole  Pi  of  any  line  h  can  therefore  be  constructed  as  the  intersec- 
tion of  the  polars  of  any  two  points  of  h  ;  this  is  of  advantage  when  the 
line  h  does  not  meet  the  circle.  And  the  polar  h  of  any  point  Pi  can  be 
constructed  as  the  line  joining  the  poles  of  any  two  lines  through  Pi ;  this 
is  of  advantage  when  the  point  Pi  lies  inside  the  circle. 

EXERCISES 

1.  Find  the  equation  of  the  polar  of  the  given  point  with  respect  to 
the  given  circle  and  sketch  if  possible : 

(a)  (4,  7),x2  +  ?/2^8. 

(6)  (0,  0),x2  +  ?/2-3x-4  =  0. 

(c)  (2,  l),x2  +  «/2_4x-2?/+l=0. 

(rZ)   (2,  -3),  x2  + 2/2+ 3a; +10?/+ 2  =  0. 

2.  Find  the  pole  of  the  given  line  with  respect  to  the  given  circle  and 
sketch  if  possible  : 

(a)  X  +  2  y  -  20  =  0,  a;2  +  y/2  =  20. 
(6)  X  +  ?/  +  1  =  0,  x2  +  ?/2  =  4. 

(c)  4  X  -  ?/  =  19,  x2  +  y2  =  25. 

(d)  Ax  +  By  +  C  =  0,  x2  +  ?/2  =  r2. 

(e)  2/  =  mx  +  6,  x2  +  i/2  =  r^. 

3.  Find  the  pole  of  the  line  joining  the  points  (20,  0)  and  (0,  10), 
with  respect  to  the  circle  x^  +  y^  =  25. 

4.  Find  the  tangent  to  the  circle  x2+«/2-10x+4  2/+9=0  at  (7,  -  6). 

5.  Find  the  intersection  of  the  tangents  to  the  circle  2  x2  +  2  y^—  15  x 
+  y  —  28  =  0  at  the  points  (3,  5)  and  (0,  —  4) . 

6.  Find  the  tangents  to  the  circle  x2  +  ]/2  —  6x  —  10  2/  +  2  =  0  that 
pass  through  the  point  (3,  —  3) . 

7.  Find  the  tangents  to  the  circle  x^  +  y2  _  3  x  +  y  —  10  =  0  that  pass 
through  the  point  (—  f,  —  V")- 

8.  Show  that  the  distances  of  two  points  from  the  center  of  a  circle 
are  proportional  to  the  distances  of  each  from  the  polar  of  the  other. 

9.  Show  analytically  that  if  two  points  are  given  such  that  the  polar 
of  one  point  passes  through  the  second  point,  then  the  polar  of  the  second 
point  passes  through  the  first  point. 

10.  Find  the  poles  of  the  lines  x  -  y  -S  =  0  and  x  +  y  +  S  =  0  with 
respect  to  the  circle  x2  +  2/-  _  6  x  +  4  y  +  3  =  0. 


106 


PLANE  ANALYTIC  GEOMETRY  [VI,  §  97 


If  in  the  left-hand  member  of  the  equa- 


97.   Power  of  a  Point. 

tion  of  the  circle 

we  substitute  for  x  and  y  the  coordinates  xi ,  ?/i  of  a  point  Pi  not  on  the 
circle  (Fig.  41),  the  expression  (xi  —  hy -\- (yi  —  k)^  —  r'^  is  different 
from  zero.     Its  value  is  called  the  power  y 

of  the  point  Pi  (xi ,  y{)  with  respect  to 
the  circle.  As  (x\  —  h)'^  +  {y\  —  k)^  is 
the  square  of  the  distance  PiC  =  d  be- 
tween the  point  Pi  {xi ,  yi)  and  the 
center  C(h,  k),  the  power  of  the  point 
Pi  (iCi,  yi)  with  respect  to  the  circle  is 
cP  —  r^;  and  this  is  positive  for  points 
without  the  circle  (d>r),  zero  for  points  Fig.  41 

on  the  circle  (d  =  r),  and  negative  for  points  within  the  circle  (d<ir). 
If  the  point  lies  without  the  circle,  its  power  has  a  simple  interpretation  ; 
it  is  the  square  of  the  segment  PiT  =  t  of  the  tangent  drawn  from  Pi  to 
the  circle : 


«2=(?2 


(^i  -  hy  -H  (yi  -  ky  -  r2. 


Hence  the  length  t  of  the  tangent  that  can  be  drawn  from  an  outside 
point  Pi  (a^i ,  yi)  to  a  circle  x'^  +  y'^ -\-  ax  -\- hy  -\-  c  =  ()  i^  given  by- 
fa  =  xi2  +  y{^  +  axi  +  hyi  +  c. 

Notice  that  the  coefficients  of  x^  and  y^  must  be  1.     Compare  the  similar 
case  of  the  distance  of  a  point  from  a  line  (§  56). 

98.    Radical  Axis.     The  locus  of  a  point  whose  powers  with  respect 

to  any  two  circles 

x2  -}-  2/2  +  axx  +  hiy  +  ci  =  0, 

a;2  +  y2  +  a^x  +  h^y  +  ca  =  0, 
are  equal  is  given  by  the  equation 

a;2  +  y2  +  a^x  +  hiy  +  ci  =  x^-\-y'^  +  a^x  -f  b^y  +  cs, 
which  reduces  to 

(ai  —  a2)x  +  (&i  —  h2)y  +  (t'l  —  ci)  -  0. 
This  locus  is  therefore  a  straight  line  ;  it  is  called  the  radical  axis  of  the 
two  circles.    It  always  exists  unless  ai  =  a<i  and  hi  =  ?)2,  i-e-  unless  the 
circles  are  concentric.  -  • 


VI,  §  99]     THE  CIRCLE.    QUADRATIC  EQUATIONS     107 

Three  circles  taken  in  pairs  have  three  radical  axes  which  pass  through 
a  common  point,  called  the  radical  center.    For,  if  the  equation  of  the 

third  circle  is 

x2  +  y2  +  asx  +  hy  +  C3  =  0, 

the  equations  of  the  radical  axes  will  be 

(a2  -  as)x  +  (62  -  b3)y  +  (C2  -  C3)  =  0, 
(as  -  ai)x  4- (&3  -  &i)y  +  (C3  -  ci)  =  0, 
(ai  -  a2)x  +  (61  -  b2)y  +  (ci  -  C2)  =  0. 
These  lines  intersect  in  a  point,  since  the  determinant  of  the  coefficients 
in  these  equations  is  equal  to  zero  (Ex.  10,  p.  57). 

99.  Family  of  Circles.     The  equation 

(8)  (a;2  4-  2/2  +  a,x  +  b^  +  c,)  +  k^x""  -\-y'-\-  a,x  +  6^  +  Cg)  =  0 

represents  a  family,  or  pencil,  of  circles  each  of  which  passes 
through  the  points  of  intersection  of  the  circles 

(9)  i«2  +  ^2^aiaj  +  &i2/+Ci  =  0, 
and 

(10)  a;2  +  2/2  +  a^x  +  h^  +  c^  =  (), 

if  these  circles  intersect.     For,  the  equation  (8)  written  in  the 
form 
(1  +  k)x2  +  (1  _^  ^^^y2  _,.  (ct^  ^  ^a2)x  +  (61  +  Kh^)y  +  Ci  +  KC2  =  0 

represents  a  circle  for  every  value  of  k  except  k  =  —  1,  as  the 
coefficients  of  x^  and  y"^  are  equal  and  there  is  no  xy-iQvm  (§  79). 
Each  one  of  the  circles  (8)  passes  through  the  common  points 
of  the  circles  (9)  and  (10)  if  they  have  any,  since  the  equation 
(8)  is  satisfied  by  the  coordinates  of  those  points  which  satisfy 
both  (9)  and  (10).  Compare  §  b^.  The  constant  k  is  called  the 
parameter  of  the  family. 

In  the  special  case  when  k  —  —  1,  the  equation  i§  of  the  first 
degree  and  hence  represents  a  line,  viz.  the  radical  axis  (§  98) 
of  the  two  circles  (9),  (10).  If  the  circles  intersect,  the  radical 
axis  contains  their  common  chord. 


108  PLANE  ANALYTIC  GEOMETRY  [VI,  §  99 

EXERCISES 

1.  Find  the  powers  of  the  following  points  with  respect  to  the  circle 
aj2  -\-y'2  —  Sx—2y=0  and  thus  determine  their  positions  relative  to  the 
circle:    (2,0),  (0,0),  (0,  -4),  (3,2). 

2.  What  is  the  length  of  the  tangent  to  the  circle :  (a)  x^ -i-  y'^  +  ax 
+  by-\-c  =  0  from  the  point  (0,  0),  (6)  {x  -  2)2  + (:«  _  3)2  -  1  =  0  from 
the  point  (4,  4)  ? 

3.  By  §  97,  t^=:d-^  —  r^=(d-{-r)(d-r);  interpret  this  relation 
geometrically. 

4.  Find  the  radical  axis  of  the  circles  x^  -\-  y^  +  ax+  by  +  c  =  0  and 
x^  +  y^  +  bx  -\-  ay  +  c  =  0  and  the  length  of  the  common  chord. 

5.  Find  the  radical  center  of  the  circles  x^-\-y^  —  Sx  +  'iy  —  7=0, 
ic2  +  ?/2  _  16,  2(a;2  -I-  ?/2)  _f.  6  X  +  1  =  0.  Sketch  the  circles  and  their  radi- 
cal axes. 

6.  Find  the  circle  that  passes  through  the  intersections  of  the  circles 
a;2  _f.  ^2  _|_  5  3j  _  0  and  x^  -\-  y'^  +  x  —  2  y  —  5  =  0,  and  (a)  passes  through 
the  point  (—5,  6),  (h)  has  its  center  on  the  line  4x  —  2y  —  l5  =  0, 
(c)  has  the  radius  5. 

7.  Sketch  the  family  of  circles  x^  +  y^  -  6  y  +  k{x^  +  ^/^  +  3  ?/)  =  0. 

8.  What  family  of  circles  does  the  equation  Ax  -{■  By  +  O  +  k(x^ 
+  y^  -\-  ax  -{■  by  -\-  c)  =  0  represent  ? 

9.  Find  the  family  of  curves  inverse  to  the  family  of  lines  y  =  mx  +  6; 
(a)  with  m  constant  and  b  variable,  (b)  with  m  variable  and  b  constant. 
Draw  sketches  for  each  case. 

10.  Show  that  a  circle  can  be  drawn  orthogonal  to  three  circles,  pro- 
vided their  centers  are  not  in  a  straight  line. 

11.  Find  the  locus  of  a  point  whose  power  with  respect  to  the  circle 
2  .^2  -f  2  ?/2  —  5  X  +  11  y  —  6  =  0  is  equal  to  the  square  of  its  distance  from 
the  origin.     Sketch. 

12.  Show  that  the  locus  of  a  point  for  which  the  sum  of  the  squares  of 
its  distances  from  the  four  sides  of  a  square  is  constant,  is  a  circle.  For 
what  value  of  the  constant  is  the  circle  real  ?  For  what  value  is  it  the 
inscribed  circle  ? 


VI,  §  99]     THE  CIRCLE.    QUADRATIC  EQUATIONS     109 

13.  Find  the  locus  of  a  point  if  the  sum  of  the  squares  of  its  distances 
from  the  sides  of  an  equilateral  triangle  of  side  2  a  is  constant. 

14.  Show  that  the  circle  through  the  points  (2,  4),  (—  1,  2),  (3,  0)  is 
orthogonal  to  the  circle  which  is  the  locus  of  a  point  the  ratio  of  whose 
distances  from  the  points  (2,  4)  and  (—  1,  2)  is  3.     Sketch. 

15.  Show  that  the  circles  through  two  fixed  points,  say  (-a,  0), 
(a,  0),  form  a  family  like  that  of  Ex.  8. 

16.  The  locus  of  a  point  whose  distances  from  the  fixed  points  (—a,  0), 
(a,  0)  are  in  the  constant  ratio  k  (:^  1)  is  the  circle 

x2  +  2/2  4-  2^-±-^ax  +  a-  =  0. 

1  —  k2 

Compare  Ex.  9,  p.  90.     Show  that,  whatever  k(:^  1),  this  circle  inter- 
sects every  circle  of  the  family  of  Ex.  15  at  right  angles. 

Parameters,     in  problems  on  loci  it  is  often  convenient  to  express 
the  coordinates  x,  y  of  the  point  describing  the  locus  in  terms  of  a  third 
variable  and  then  to  eliminate  this  variable.     Thus,  for  any  point  on  a 
circle  of  radius  a  about  the  origin  we  have  evidently 
(a)  X  =  a  cos  0,        y  =  a  sin  <f> ; 

eliminating  <p  by  squaring  and  adding  we  find 

^•2  ^  y2  -  ^2. 

The  variable  <p  is  called  the  parameter;    the  equations   (a)   are  the 
parameter  equations  of  a  circle  about  the  origin. 

17.  The  ends  ^,  jB  of  a  straight  rod  of  length  2  a  move  along  two  per- 
pendicular lines  ;  find  the  locus  of  the  midpoint  of  AB. 

18.  One  end  vl  of  a  straight  rod  of  length  a  describes  a  circle  of  radius  a 
and  center  O,  while  the  other  end  B  moves  along  a  line  through  0.  Taking 
this  line  as  axis  Ox  and  0  as  origin,  find  the  locus  of  the  intersection  of 
OA  (produced)  with  the  perpendicular  to  the  axis  Ox  through  B. 

19.  Four  rods  are  jointed  so  as  to  form  a  parallelogram  ;  if  one  side  is 
fixed,  find  the  path  described  by  any  point  rigidly  connected  with  the  op- 
posite side. 

20.  An  inversor  is  any  mechanism  for  describing  the  inverse  of  a  given 
curve.  Peaucellier's  cell  consists  of  a  linked  rhombus  APBP'  attached 
by  means  of  two  equal  links  OA,  OB  to  a  fixed  point  0.  Show  that  this 
linkage  is  an  inversor,  with  O  as  center. 


CHAPTER   VII 

COMPLEX   NUMBERS 

PART   I.     THE   VARIOUS   KINDS   OF  NUMBERS 

100.  Introduction.  The  process  of  finding  the  points  of  in- 
tersection of  a  line  and  a  circle  (§  86)  involves  the  solution  of 
a  quadratic  equation.  The  solution  of  such  a  quadratic  equa- 
tion may  involve  the  square  root  of  a  negative  number.  Thus 
the  roots  of  ic^  —  2a;  +  3  =  0  are  x  =  l  ±^—2. 

The  square  root,  or  in  fact  any  even  root,  of  a  negative  num- 
ber is  called  201  imaginary  number;  and  an  expression  of  the 
form  a  +  V—  6  in  which  a  is  any  real  number  and  b  any  posi- 
tive real  number  is  called  a  complex  number. 

We  shall  first  recall  briefly  the  successive  steps  by  which,  in 
elementary  algebra,  we  are  led  from  the  positive  integers  to 
other  kinds  of  numbers. 

101.  Fundamental  Laws  of  Algebra.  The  so-called  natural 
numbers,  or  positive  integers  1,  2,  3,  4,  •  •  •  form  a  class  of 
things  for  which  the  operations  of  addition  and  midtiplication 
have  a  clear  and  well-known  meaning.  These  operations  are 
governed  by  the  following  laws  : 

(a)  the  commutative  law  for  addition  and  for  multiplication : 

a-\-b  =  b  i-  a,  ab  =  ba\ 

(p)  the  associative  law  for  addition  and  for  multiplication : 

(a  +  6)  -f  c  =  a  +  (6  -f  c),  {ab)c  =  a{bc) ; 

(c)  the  distributive  law,  connecting  addition  and  multiplication  : 

{a -\- b)  c  =  ac -\-  be,  a(b  -\-c)  =  ab-\-  ac. 

110 


VII,  §103]  COMPLEX  NUMBERS  111 

102.  Inverse  Operations.  The  result  obtained  by  adding 
or  multiplying  any  two  or  more  positive  integers  is  alwaj^s 
again  a  positive  integer. 

This  is  not  true  for  the  so-called  inverse  operations :  subtrao- 
tion,  the  inverse  of  addition,  and  division,  the  inverse  of  multi- 
plication. To  make  these  inverse  operations  always  possible 
the  domain  of  positive  integers  is  extended  by  introducing : 

(a)  the  negative  numbers  and  the  number  zero  ; 

(6)  the  (positive  and  negative)  rational  fractions. 

The  relation  between  these  various  kinds  of  numbers  is  best 

understood     by     imagining        -ji     -si     -i\       \o      \i      [g      p ^ 

the  positive  integers  repre-  Fig.  42 

sented  by  equidistant  points  on  a  line,  or  rather  by  the  distances 

of  these  points  from  a  common  origin  O  (Fig.  42). 

Negative  numbers  are  then  represented  by  equidistant  points 
on  the  opposite  side  of  the  origin;  zero  is  represented  by  the 
origin ;  and  fractions  correspond  to  intermediate  points. 

103.  Rational  Numbers.  The  positive  and  negative  inte- 
gers, the  rational  fractions,  and  zero,  form  the  domain  of  ra- 
tional numbers.  By  adopting  the  well-known  rules  of  signs  the 
operations  of  addition  and  multiplication  and  their  inverses, 
subtraction  and  division,  can  be  extended  to  these  rational  num- 
bers ;  and  all  four  of  these  operations,  with  the  single  exception 
of  division  by  zero,  can  be  shown  to  be  always  possible  in  the 
domain  of  rational  numbers,  so  that  any  finite  number  of  such 
operations  performed  with  a  finite  number  of  rational  numbers 
produces  again  a  rational  number. 

In  the  domain  of  positive  integers  such  linear  equations  as 
a;-f-7  =  0,  5a;  —  3  =  0  cannot  be  solved.  But  in  the  domain  of 
rational  numbers  the  linear  equation  ax-\-b  =  0  can  always  be 
solved  if  a  and  b  are  rational  and  a  is  not  zero. 


112  PLANE  ANALYTIC  GEOMETRY      [VII,  §  104 

104.  Laws  of  Exponents.  In  the  domain  of  positive  inte- 
gers, we  pass  from  addition  to  multiplication  by  denoting  a 
sum  of  h  terms  each  equal  to  a  by  the  symbol  ah,  called  the 
product  of  a  and  h.  Similarly,  we  may  denote  a  product  of 
b  factors  each  equal  to  a  by  the  symbol  aJ* ;  this  operation  is 
called  raising  a  to  the  bth  poicer,  or  involution.  By  this  defini- 
tion, the  symbol  a^  has  a  meaning  only  when  the  exponent  h  is 
a  positive  integer.  But  the  base  a  may  evidently  be  any 
rational  number.     The  laws  of  exponents,  or  of  indices, 

a^  '  a''  =  ap+'^,     a^  •  5^=  (a^)p,      {a^y  =  a^", 
follow  directly  from  the  definition  of  the  symbol  a*.     The  re- 
sult of  raising  any  rational  number  to  a  positive  integral  power 
is  always  a  rational  number. 

105.  The  Inverses  of  Involution.     It  should  be  observed 

that  the  symbol  a^  differs  from  the  symbols  a  +  h  and  ah  in 

not  being  commutative  (§  101)  ;  i.e.  in  general  a  and  h  cannot 

be  interchanged: 

a^  =/=  6",     if     h^  a.' 

It  follows  from  this  fact  that  while  addition  and  multiplication 

have  each  but  one  inverse  operation,  involution  has  two : 

(a)  If  in  the  relation 

a^  =  G 

h  and  c  are  regarded  as  known,  the  operation  of  finding  a  is 
called  extracting  the  hth  root  of  c,  or  evolution,  and  is  expressed 
in  the  form 

a  =  Vc. 
(h)  If  in  the  same  relation  a  and  c  are  regarded  as  known, 
the  operation  of  finding  h  is  called  taking  the  logarithm  of  c  to 
the  hase  a  and  is  indicated  by 

b  =  log„  c. 
Logarithms  will  be  discussed  in  Chapter  XII ;  for  the  present 
we  shall  consider  only  the  former  inverse  operation. 


VII,  §  107]  COMPLEX  NUMBERS  113 

106.  Irrational  Numbers.  Even  when  a,  h,  and  therefore  c 
are  positive  integers,  the  extraction  of  roots  is  often  impossible, 
not  only  in  the  domain  of  positive  integers,  bat  even  in  the 
domain  of  rational  numbers.  Thus,  in  so  simple  a  case  as 
6  =  2,  c  =  2,  we  find  that  a  =  V2  is  not  a  rational  number,  i  e. 
it  is  not  the  quotient  of  any  two  integers,  however  large.  For, 
suppose  that  V2  =  h/k,  where  h  and  k  are  integers  and  the  ra- 
tional fraction  h/k  is  reduced  to  its  lowest  terms ;  then  squaring 
both  sides,  we  find  2  =  h^/k^.  But  the  rational  fraction  h^/k"^ 
is  also  reduced  to  its  lowest  terms  and  consequently  cannot 
be  equal  to  the  integer  2. 

We  are  thus  led  to  a  new  extension  of  the  number  system 
by  including  the  results  of  evolution :  any  root  of  a  rational 
number  that  is  not  a  rational  number  is  called  an  irrational 
number.  The  rational  and  irrational  numbers  together  form 
the  domain  of  real  numbers. 

If  numbers  are  represented  by  points  on  a  line  as  in  §  102, 
the  number  V2  has  a  single  definite  point  corresponding  to  it 
on  the  line ;  for,  the  segment  representing  it  can  be  found  as 
the  hypotenuse  of  a  right  triangle  whose  sides  have  the  length  1. 
It  can  be  shown  that  a  single  definite  point  corresponds  to 
any  given  irrational  number. 

It  thus  appears  that  although  the  rational  numbers,  "  crowd 
the  line,"  i.e.  although  between  any  two  rational  numbers,  how- 
ever close,  we  can  insert  other  rational  numbers,  they  do  not 
"  fill "  the  line ;  i.e.  there  are  points  on  the  line  that  cannot 
be  represented  exactly  by  rational  numbers. 

107.  Extension  of  Laws.  A  rigorous  definition  and  dis- 
cussion of  irrational  numbers  requires  somewhat  long  and  com- 
plicated developments.  It  will  here  suffice  to  state  the  result 
that  irrational  numbers  are  subject  to  the  same  rules  of  operation 
as  are  rational  numbers. 


114  PLANE  ANALYTIC  GEOMETRY      [VII,  §  107 

The  fundamental  laws  of  addition  and  multiplication  (§  101) 
hold  therefore  for  all  real  numbers,  and  so  do  the  laws  of  signs 
of  elementary  algebra.  As  regards  the  laws  of  exponents 
(§104),  they  can  be  shown  to  hold  when  the  bases  are  any  real 
numbers.  Moreover,  it  can  be  shown  that  the  symbol  a^  has 
a  definite  meaning  even  when  the  exponent  h  is  any  real  num- 
ber, and  that  the  laws  of  exponents  hold  for  such  powers,  pro- 
vided only  that  the  bases  are  positive.  It  is  known  from 
elementary  algebra  how  this  can  be  done  for  rational  exponents 
by  defining  the  symbols  a°  and  a~"*  as 

a«  =  l,      a-^  =  —: 

a*" 

and  it  is  shown  in  the  theory  of  irrational  numbers  that  the 
latter  definition  can  be  used  even  when  m  is  irrational. 

Thus  the  laws  of  exponents  (§  104)  hold  for  any  real  ex- 
ponents provided  the  bases  are  positive. 

108.  Measurement.  Historically,  the  gradual  introduction 
of  rational  fractions,  of  negative  numbers,  of  irrational  num- 
bers, was  determined  very  largely  by  the  ajyplications  of  arith- 
metic and  algebra.  Any  magnitude  that  can  be  subdivided 
indefinitely  into  parts  of  the  same  kind  as  the  whole,  and 
hence  can  be  "measured,"  leads  naturally  to  the  idea  of  the 
fraction.  Magnitudes  that  can  be  measured  in  two  opposite 
senses,  like  the  distance  along  a  line,  the  height  of  the  ther- 
mometer above  and  below  the  zero  point,  credit  and  debit,  the 
height  of  the  water  level  above  or  below  a  fixed  point,  suggest 
the  idea  of  negative  numbers.  The  incommensurable  magnitudes 
that  occur  frequently  in  geometry  lead  to  the  introduction  of 
irrational  numbers.  One  of  the  principal  advantages  of  algebra 
consists  in  the  remarkable  fact  that  all  these  different  kinds  of 
numbers  are  subject  to  the  same  simple  laws  of  operation. 


VII,  §  110]  COMPLEX  NUMBERS  115 

109.  Imaginary  Numbers.  As  mentioned  in  §  107,  there  is 
still  a  restriction,  in  the  domain  of  real  (i.e.  rational  and 
irrational)  numbers,  to  the  use  of  the  laws  of  exponents  (§  104)  : 
the  square  root  of  a  negative  number  has  no  meaning  in  this 
domain. 


Thus,  V—  2  is  not  a  real  number ;  for,  by  the  definition  of 
the  square  root,  the  square  of  V—  2  is  —  2 ;  but  there  exists 
no  real  number  whose  square  is  —  2.  In  other  words,  such 
simple  equations  as  x^  +  2  =  0,  a;^  —  2  a;  +  3  =  0  have  no  real 
solutions.  It  has  therefore  been  found  of  advantage  to  give  one 
further  extension  to  the  meaning  of  the  term  "  number,"  by 
including  the  even  roots  of  negative  numbers,  under  the  name 
of  imaginary  numbers. 

110.  The  Imaginary  Unit.  Any  even  root  of  a  negative 
(rational  or  irrational)  number  is  defined  as  an  imaginary 
number.  Every  such  number  can  be  reduced  to  the  form 
±  V—  a,  where  a  is  positive.  It  is  customary  to  denote  V  — 1 
by  the  letter  /  and  call  it  the  imaginary  unit.  Any  imaginary 
number  ±  V—  a  can  therefore  be  written  in  the  form 

±  V  —  a  =  ±  Va  i ; 
that  is,  every  imaginary  number  is  a  real  multiple  of  the  imag- 
inary unit  I.     Notice  that  as  i  =  V—  1  we  always  have 

1-2  =  -  1. 
The  algebraic  sum  of  a  real  number  and  an  imaginary  num- 
ber, i.e.  the  expression  a  +  bi  where  a  and  b  are  real,  is  called 
a  complex  number.  Notice  that  the  domain  of  complex  num- 
bers includes  both  real  and  imaginary  numbers.  For,  the 
complex  number  a  +  bi  is  real  in  the  particular  case  when 
6  =  0,  it  is  an  imaginary  number  if  a  =  0.  The  great  advan- 
tage of  complex  numbers  lies  in  the  fact  that  all  the  seven 
fundamental  operations  of.  algebra  (viz.  addition,  subtraction, 


116  PLANE  ANALYTIC  GEOMETRY      [VII,  §  110 

multiplication,  division,  involution,  evolution,  and  logarithmi- 
zation),  with  the  single  exception  of  division  by  zero,  can  be 
performed  on  complex  numbers,  the  result  being  always  a 
complex  number ;  i.e.  if  we  denote  by  a,  (3  any  two  complex 
numbers,  then  a  -\- 13,  a  —  (3,  a(3,  a/^,  a^,  -v/cc,  log^  a  can  all  be 
expressed  in  the  form  a  +  bi.  It  can  then  be  shown  that  every 
algebraic  equation  of  the  nth  degree  has  7i  complex  roots. 

111.  Imaginary  Values  in  Analytic  Geometry,  in  elemen- 
tary analytic  geometry  we  are  concerned  with  "real"  points  and  lines, 
i.e.  with  points  whose  coordinates  are  real  and  with  lines  whose  equations 
have  real  coefficients.  But  it  should  be  observed  that  points  with  com- 
plex coordinates  may  lie  on  real  lines  and  that  lines  with  complex  coeflQ- 
cients  may  contain  real  points.  Thus,  the  coordinates  of  the  point 
(2  +  3  i,  1  —  2i)  satisfy  the  equation  of  the  real  line  2x  +  Sy— 7  =  0, 
and  the  equation  (I  +  2  i)x  —  {2  -^  S  i)  y  -{•  1  =  0  is  satisfied  by  the  point 
(3,  2).  Calculations  with  imaginary  points  and  lines  may  therefore  lead 
to  results  about  real  points  and  lines. 

A  rather  striking  example  is  afforded  by  the  the  theory  of  poles  and 
polars  with  respect  to  the  circle.  We  have  seen  (§§  93-95)  that  with 
respect  to  a  given  circle  every  line  of  the  plane  (excepting  those  through 
the  center)  has  a  real  pole  and  every  point  (excepting  the  center)  has  a 
real  polar.  If  the  line  I  intersects  the  circle  in  two  points  §i ,  ^2 »  its 
pole  P  can  be  found  as  the  intersection  of  the  tangents  at  Qi,  Q2.  If  the 
line  I  does  not  intersect  the  circle,  this  geometrical  construction  is  im- 
possible. But  the  analytic  process  of  finding  the  points  of  intersection  of 
the  line  I  with  the  circle  can  be  carried  through.  The  coordinates  of  the 
points  of  intersection  will  be  imaginary  ;  and  hence  the  equations  of  the 
tangents  at  these  points  will  have  imaginary  coefficients.  But  the  point 
of  intersection  of  these  imaginary  lines  will  be  a  real  point  ;  viz.  the  pole 
P  of  the  line  I  and  its  real  coordinates  can  be  found  in  this  way. 

Thus  to  find  the  pole  of  the  line  y  =  2  with  respect  to  the  circle 
x^-^y^  =  l  we  obtain  the  imaginary  points  of  intersection  (VSi,  2)  and 
(—  V3i,  2)  ;  the  imaginary  tangents  at  these  points  are  therefore: 
VSix  -f-  2  y  =  1,  —  VS  ix  +  2y  =  1;  these  imaginary  lines  intersect  in  the 
real  point  (0,  ^);  it  is  easy  to  show  that  this  is  the  required  pole. 


Vn,  §  113]  COMPLEX  NUMBERS  117 

PART  II.     GEOMETRIC  INTERPRETATION  OF 
COMPLEX  NUMBERS 

112.  Representation  of  Imaginaries.  The  meaning  of  com- 
plex numbers  will  best  be  understood  from  their  graphical 
representation. 

We  have  seen  (§  102)  that  every  real  number  a  can  be  repre- 
sented by  a  point  ^  on  a  straight  line  on  which  an  origin  0 
and  a  positive  sense  have  been  selected.  _    . 

We  shall  call  this  line  (Fig.  43)  the 
axis  of  real  numbers,  or  briefly  the  real 

axis.  I  I       o      \   BealAxis 

To  represent  the  imaginary  numbers 
we  draw  an  axis  through  O  at  right 
angles  to  the  real  axis  and  call  it  the 
axis  of  imaginary  numbers,  or  briefly  ^^'    ' 

the  imaginary  axis.  The  point  A'  on  this  axis,  at  the  distance 
OA'  —  a  from  the  origin,  can  then  be  taken  as  representing 
the  imaginary  number  ai. 

113.  Representation  by  Rotation.  This  representation  is 
also  suggested  by  the  fundamental  rule  for  dealing  with  im- 
aginary numbers  that  i*  =  —  1.  For,  if  a  be  any  real  number 
and  A  its  representative  point  on  the  real  axis,  the  real  num- 
ber —  a  has  its  representative  point  A'  situated  symmetrically 
to  A  with  respect  to  0  on  the  real  axis ;  in  other  words,  the 
segment  OA'  which  represents  —  a  can  be  regarded  as  ob- 
tained from  the  segment  OA  that  rejjresents  a  by  turning  OA 
through  two  right  angles  about  0.  Thus  the  factor  —  1  =  t^ 
applied  to  the  number  a,  or  rather  to  the  segment  OA,  turns 
it  about  0  through  two  right  angles.  This  suggests  the  idea 
that  the  factor  V—  1  =  ?*,  applied  to  a,  may  be  interpreted  as 


118  PLANE  ANALYTIC  GEOMETRY      [VII,  §  113 

turning  the  segment  OA  through  one  right  angle  in  the  counter- 
clockwise sense  so  as  to  make  it  take  the  position  OA'.  Indeed, 
if  the  factor  i  be  now  applied  to  ai,  i.e.  to  the  segment  OA',  it 
will  turn  OA'  into  OA"  and  produce  ai^  =  —  a. 

Turning  OA"  counterclockwise  through  a  right  angle,  we 
obtain  the  point  A'"  on  the  imaginary  axis  which  represents 
ai^  =  —  ai;  and  finally,  turning  OA"'  counterclockwise  through 
a  right  angle  we  regain  the  starting  point  A  which  represents 
ai^  =  a. 

114.  Representation  of  Complex  Numbers.  A  complex 
number,  i.e.  an  expression  of  the  form 

z=x  +  yi, 

where   x,  y  are  real  numbers  while  i  is   the  imaginary  unit 

V—  1,  is  fully  determined  by  the  two  real  numbers  x  and  y, 

provided  we  know  which  of  these  is  to  be  the  real  part.     If 

we  take  the  real  axis  as  axis  Ox,  the  imaginary  axis  as  axis 

Oy,  of  a  rectangular  coordinate  system 

(Fig.  44),  the  numbers  x,  y  determine 

a  definite  point  of  the  plane,  and  only 

one.     This  point  P{x,  y)  can  therefore 

be  taken  as  representative  of  the  com-    ^  <?       ^ 

plex  number  z—x-\-  yi.  ^ict.  44 

This  representation  also  agrees  with  the  idea  (§  113)  that  the 
factor  i  turns  through  a  right  angle.  For  if  we  lay  off  on  the 
real  axis,  or  axis  Ox,  OQ  =  x,  and  on  the  same  axis  QR  =  y 
we  obtain  OR  =  OQ  +  QR  =  x-\-y;  and  if  we  turn  QR  about 
Q  through  a  right  angle  into  QP  we  obtain  x  +  yi  and  reach 
the  point  P. 

To  every  complex  number  z  =  x  -\-  yi  thus  corresiDonds  one  and 
only  one  point  P(x,  tj)  ;  to  every  point  P(x,  y)  of  the  plane  cor- 
responds one  and  only  one  complex  number  z  =  x  -{■  yi. 


VII,  §116]  COMPLEX  NUMBERS  119 

The  real  numbers,  and  only  these,  have  their  representative 
points  on  the  axis  Ox-^  the  imaginary  numbers  have  theirs  on 
the  axis  Oy.  The  origin  (0,  0)  represents  the  complex  num- 
ber 0-{-  iO  =  0. 

115.  Correspondence  of  Complex  Numbers  to  Vectors.    It 

should  be  recalled  that  strictly  speaking  (§  102)  a  real  number  x 
is  represented,  not  by  a  point  A  of  the  real  axis,  but  by  the 
segment  OA  =  x.  Similarly  the  complex  number  z=:x-{-yi  is 
represented,  strictly  speaking,  not  by  the  point  P  (Fig.  44),  but 
rather  by  the  radius  vector  OP,  taken  with  a  definite  direction 
and  sense.  Thus  the  complex  number  z  =  x-\-yi  represents  a 
vector  (see  §§  19-20),  whose  rectangular  components  are  x 
and  y.  It  will  be  shown  below  that  the  addition  and  subtrac- 
tion of  complex  numbers  follow  exactly  the  laws  of  the  com- 
position of  (concurrent)  forces,  velocities,  translations,  etc.,  in 
the  same  plane. 

116.  Equality  of  Complex  Numbers.  Two  complex  num- 
bers Z]^  =  x^-\-  y^i  and  Z2  =  x^-{-  y^i  are  called  equal,  if,  and  only 
if,  their  representative  points  coincide,  i.e.  z^  =  z^  if 

x^  =  X2  and  yi  =  y^, 

just  as  two  forces  are  equal  only  when  their  rectangular  com- 
ponents are  equal  respectively. 

If  we  apply  the  ordinary  rules  of  algebra  to  the  equation 

^i  +  2/i^*  =  ^2  4-  yii 
we  obtain 

Xi-X2  =  (2/2  -  yi)i- 

Now  the  real  number  x^  —  x^  cannot  be  equal  to  the  imaginary 
number  {y^  —  y^i  unless  ajj  —  x^^O  and  2/2  —  2/i  =  0 ;  whence 
again  we  find  a^i  =  x^,  y^  =  y^. 

It  follows  in  particular  that  the  complex  number  z  =  x -{-  yi 
is  zero  if,  and  only  if,  a;  =  0  and  y  =  0. 


120 


PLANE  ANALYTIC  GEOMETRY      [VII,  §  116 


EXERCISES 

1.  Locate  the  points  which  represent  the  following  complex  numbers  : 
(a)  4-3  i.  (6)  2  i.  (c)   -  1  -  i.  (d)  4. 
(e)  A  +  .li.            if)  f-li.              {g)   -10-ti.              {h)   -ii. 

2.  Find  the  values  of  m  and  n  in  the  following  equations  : 

(a)   {m  -  n)  +  (to  +  w  -  2)z=  0.        (6)   (m2+w2-25)  +  (w-w-l)i=0. 
(c)  w  +  ni  =  3  —  2 1.  ((?)  mm  =  m^  -  w^  -j-  4  i. 

3.  Show  that 

(a)   l3  =_i,  (6)   1-5  3=  i9^  (c)   I'C  +  i^  =  0,  (c?)   1-4  -  l6  =  2. 

4.  Show  that  the  following  relations  are  true,  n  being  any  positive 
integer : 

(a)  i^'  =  \.  (6)  i^+^=-i.  (c)  i*«-?>+2  =  2. 

5.  Show  that 

(«)  K—  1  +  VS  i)  is  a  cube  root  of  1, 
{h)  J (4-  1  —  y/Zi)  is  a  cube  root  of  —  1. 

117.  Addition  of  Complex  Numbers.  The  sum  of  tivo 
complex  numbers  Zi  =  Xi-\-  y^i  and  Z2  —  X2  +  y^i  is  defined  as  the 
complex  number  z=  (xi-{-  x^)  -\-  (2/1+2/2)*  j  ii^  other  words,  if  (Fig. 
45)  Pi  is  the  point  that  represents  Zi  and  P^  the  point  that  rep- 
resents Z2,  then  the  point  P  that  repre- 
sents the  sum  z  =  Zi-^Z2  has  for  its  ab- 
scissa the  sum  of  the  abscissas  of  Pj 
and  P2  and  for  its  ordinate  the  sum  of 
the  ordinates  of  Pj  and  P.^.  It  appears 
from  the  figure  that  this  point  P  is  the 
fourth  vertex  of  the  parallelogram  of 
which  the  other  three  vertices  are  the  origin  0  and  the  points 

P„P2. 

118.  Analogy  to  Parallelogram  Law  of  Vectors.     By  com- 
paring §§  19,  20  it  will  be  clear  that  the  addition  of  two  com- 


y 

P 

R^--^ 

'/^\ 

«!   1    . 

~0 

""^Qz 

<4   9 

Fig. 

45 

VII,  §  119] 


COMPLEX  NUMBERS 


121 


plex  numbers  consists  in  finding  the  resultant  OP  of  their 
representative  vectors  OP^,  OPj..  The  vectors  may  be  thought 
of  as  forces,  velocities,  translations,  etc.  In  the  case  of  trans- 
lations this  composition  of  two  successive  translations  into  a 
single  equivalent  translation  is  particularly  obvious. 

While  a  real  number  «=  OQ  represents  a  translation  along 
the  axis  Ox*,  an  imaginary  number  yi  a  translation  along  the 
axis  O2/,  a  complex  number  z  —  x-\-yi  can  be  interpreted  as 
representing  a  translation  OP  in  any  direction  (Fig.  44).  The 
succession  of  two  such  translations  %  =  a^  -|-  y^  represented  by 
OPx  (Eig.  45)  and  z.2,  =  x^-\-  y^i  represented  by  OP^  is  equivalent 
to  the  single  translation  z=  (a?,  -^x^  -f-  (y^  H-//2)*'  represented 
by  OP. 

It  follows   that   the  addition  of   any  number    of    complex 
numbers    (Fig.   46)    whose 
representative   vectors   are 
OPi,  OP2,  OP3,  OP4  can  be 
effected    by    forming    the 


Fia.  46 


polygon  0PiP2'Ps'P;  the 
closing  line  OP  is  the  rep- 
resentative vector  of  the 
sum  ;  precisely  as  in  finding 
the  resultant  of  concurrent 
forces  (§  20). 

119.  Subtraction.  The  difference  of  two  complex  numbers 
2;^  =  iCi  -f-  y^i  and  22  =  ^2  +  2/2**  *^  ^6- 
fined  as  the  complex  number  z  =  (a^j 
—  ^2)  H-  (2/1  —  2/2)*'-  Its  representative 
point  P  is  found  geometrically  by 
laying  off  from  P^  (Fig.  47)  a  seg- 
ment PjP  equal  and  opposite  to 
OP2,  i.e.  equal  and  parallel  to  P2O. 


122  PLANE  ANALYTIC  GEOMETRY      [VII,  §  120 

120.  Multiplication.  The  product  of  two  complex  yiumhers 
Zi  =  Xi  +  2/l^  and  z^  =  X2  +  2/2^'  ^^  found  by  multiplying  these  two 
expressions  according  to  the  ordinary  rules  of  algebra  and  observ- 
ing that  1*2  =  —  1.     We  thus  find  : 

z^z^  =  (%  +  2/iO(^'2  +  ^20  =  ^1^2  +  ^m  +  ^22/1*  +  2/l2/2*^ 
=  {^v«2  -  2/12/2)  +  (a^i2/2  +  ^iV^h 
which  is  a  complex  number.     A  geometric  construction  will 
be  given  in  §  124. 

121.  Conjugate  Imaginaries.  Two  complex  numbers  that 
differ  only  in  the  sign  of  the  imaginary  part  are  called  con- 
jugate complex  numbers.  Thus,  the  conjugate  of  5-|-2i  is 
5  —  2  i ;  that  of  —  3  —  ^  is  —  3  +  i,  etc.  The  radii  vectores  rep- 
resenting two  conjugate  numbers  are  situated  symmetrically 
with  respect  to  the  real  axis. 

Tlie  product  of  two  conjugate  complex  numbers  is  a  real 
number;  for 

{x  +  yi){x  -  yi)  =:x'^+  y\ 

Notice  that  the  roots  of  a  quadratic  equation  are  conjugate 
complex  numbers. 

122.  Division.  To  form  the  quotient  of  two  complex  num- 
bers we  may  render  the  denominator  real,  by  multiplying  both 
numerator  and  denominator  by  the  conjugate  of  the  denomi- 
nator.    Thus : 

?i  _  ^\  +  y\i  ^  (ag  +  .ViOfe  — .VaO  _  a?ia?2— aa?/2^'  +  ^'2y\i+  ViVi 

2^2        3^2  4-2/2**         (X2^-y.2i){X2  —  y2i)  X2+y2 

=  (^^&±M^\  4-  (^2V\  —  ^y^i 

\  x,^+y,'  J     \xi  +  yi  )' 

Here  also  the  result  is  a  complex  number.    A  geometric  con- 
struction is  indicated  in  §  125. 


yil,  §  122]  COMPLEX  NUMBERS  123 


EXERCISES 

1.  Simplify  the  following   expressions  and   illustrate  by  geometric 
construction  : 

(a)   (3-60  +  (4-20.  (6)   (4  -  3i)-(2  +  i). 

(c)   (6+0  +  (3-20-(0.  (d)   (2-30-(-l+0-(3  +  50. 

(e)   (4)-(30.  (/)  (0  +  (3-2i)-(6). 

2.  Write  the  following  products  as  complex  numbers  and  locate  the 
corresponding  points  : 

(a)   (V5  +  iV6)(\/6+iV5).  (6)   (3-zV8)(V3-fiV2). 

(c)  (vrTT-vn^)2.  (d)  (Va-V^^)3. 

3.  Show  that 

.  s  l  +  2i      l-2i  ^  3    (&)   (X  +  2/0^  -  (a;  -  2/0^  =  4a;?/i. 

*^  M  +  i        1-1         ■  (c)  (ix+yiy  +  <ix-yiy  =  2(x'^  +  f)-12x^y^. 

4.  Write  the  following  quotients  as  complex  numbers  and  locate  the 
corresponding  points : 

(a) 


2  +  3i 
4-i 

(^)^-^^ 

(c)  ^-3\ 
^  ^  6  +  3i 

(1  +  0(1  +  20(1  +  30 

(0     '    . 

(/)     -  ^    . 

l  +  4i 

^  ^    -7+2i 

'^'^^3-4i 

(d) 

5.  Verify  by  geometric  construction  that  the  sum  of  two  conjugate 
complex  numbers  is  a  real  number  and  that  the  difference  is  an  imaginary 
number. 

6.  Evaluate  the  following  expressions  for  ^i  =  3  +  4  i  and  Z2  =  —  2  +  6i 
and  check  by  geometric  construction : 

(a.)  01-6.  (&)  2^2  +  3.  (c)  6-501.  (d)  Si-{-2zi. 

(e)  2i-\zi.        (/)  2-202.  {g)  1(1-^1).  {h)   -Si-z^. 

(0    01+2  02.  U)     3  01  +  02.  (fc)   01-2  02.  (l)     Zo-lZi. 

(??i)  01  +  502— 4  i.  (w)  02—^01  +  3.      (o)  5  —  01  —  02.      (p)02  — 6  — f0i. 

7.  Let  Xi  and  ri  represent  the  projections  of  a  force  Fi  on  the  axes 
of  X  and  y,  respectively,  and  X2  and  F2  those  of  a  second  force  F2.  Show, 
by  the  parallelogram  law,  that  the  projections  on  the  axes  of  the  result- 
ant (or  sum)  of  Fi  and  F2  are  Xi  +  X2  and  Yi  +  T2. 

8.  From  Ex.  7,  show  that  the  correct  results  are  obtained  if  Fi  is 
represented  by  Xi  +  Yii,    F2  by  X2  +  ¥21,  and  their  resultant  by 

Fi-{-F2=  (Xi  +  FiO  +  (X2  +  Tzi)  =  (Xi  +  X2)  +  ( Ti  +  ¥2)1. 


124 


PLANE  ANALYTIC  GEOMETRY       [VII,  §  123 


123.    Polar  Representation.      The  use  of  the  polar  coordinates  r, 
0  of  the  representative  point  P{Xj  y)  leads  to  simple 
interpretations  of  multiplication,  division,  involution, 
and  evolution. 

The  distance  OP  =  r  (Fig.  48)  is  called  the  modulus 
or  absolute  value  of  the  complex  number  ;  the  vec- 
torial angle  0  is  sometimes  called  the  argument,  phase ^ 
or  amplitude. 


Fig.  48 


Since 

we  can  write 

The  right-hand  member  of  this  equation  is  the  polar  form  of  the  complex 

number  z  =  x  -\-  yi. 


x  =  r  cos  0  and  y  =  r  &m.  0, 
z  —  X  -\-yi  =  r(cos  0  +  i  sin  0) . 


124.  Products  in  Polar  Form.     The  product  of  two  complex 

numbers  z\  =  ri(cos  0i  +  i  sin  0i)    and  02  =  »*2(cos  02  +  i  sin  02)  is 

0i2r2=?'i(cos0i+tsin0i)r2(cos02  +  isin02) 

=rir2[(cos0icos02  — sin0isin02)  +i(sin  0i  cos02+cos0i  sin  02)] 
=rir2[cos(0i  +  02)  +  isin(0i  +  02)]. 

This  shows  that  the  modulus  of  the  product  of  two  complex  numbers  is 
the  product  of  the  moduli,  the  amplitude  of  the  product  is  the  sum  of  the 
amplitudes,  of  the  factors. 

The  point  P  that  represents  the  product  of  the  complex  numbers  repre- 
sented by  the  points  Pi  and  P2  (Fig.  49)  can  be  constructed  as  follows : 
Let  Po  be  the  point  on  the  axis  Ox  at  unit  distance 
from  the  origin  0  and  draw  the  triangle  OPqPi  ; 
on  OP2  construct  the  similar  triangle  OP2P.  The 
point  P  thus  located  is  the  required  point.  For, 
by  construction  the  angle  P2OP=0i,  hence  the 
angle  PoOP=:  01  +  02-  Moreover,  as  the  triangles 
OPoPi  and  OP2P  are  similar,  their  sides  are  pro- 
portional, i.e. 

1  :  n  =  r2  :  OP,  whence  OP  =  rir2. 


Fig.  41) 


125.  Quotients  in  Polar  Form.  For  the  quotient  of  the  two 
complex  numbers  zi  =  ri(cos  0i  +  i  sin  0i)  and  02  =  r2(cos  02  +  i  sin  02) 
we  find  by  making  the  denominator  real : 


VII,  §  125] 


COMPLEX  NUMBERS 


125 


£i  _  n (cos  01  +  i  sin  0i)  _  ri(cos  <pi  +  i  sin  0i)  (cos  02  —  i  sin  02) 
«2     >'2(cos  02  +  I  sin  02)      r2(cos  02  +  i  sin  02)  (cos  02  —  i  sin  02) 
_ri     (cos  01  cos  02  4-  sin  0i  sin  02)  +  ^^(sin  0i  cos  </)2  — cos  0i  sin  ^2) 
r2  cos2  02  +  sin2  02 

=  ^  [cos  (01  -  02)  +1  sin  (01 -02)]. 
r2 

Hence  the  modulus  of  the  quotient  z  —  zi/z^  is  the 
quotient  of  the  moduli,  the  amplitude  is  the  differ- 
ence of  the  amplitudes  of  Zi  and  z^.  Evidently  the 
point  P  that  represents  the  quotient  z  =  Z1/Z2 
Fig.  50)  can  be  located  by  reversing  the  geometric 
construction  given  in  §  124;  i.e.  by  constructing 
on  the  unit  segment  OPq  the  triangle  OPqP  similar 
to  the  triangle  OP2P1. 

EXERCISES 

1.  Write  the  following  complex  numbers  in  polar  form  : 

(a)  2  +  2V3  i.        (&)  -  3  +  3  V3  i.     (c)  6-6  i.  (d)  -  5  i. 

(e)         7.  (/)  -8.  (^)  5\/3-5i.      (h)  -10-lOi. 

2.  Write  the  following  complex  numbers  in  the  form  x  +  yi: 


(a)  3(cos30°  +  isin30^). 
(c)   10(cos  I TT  +  i  sin  I  tt)  . 
(e)    V2(cosi7r +  isin^Tr). 
(g)  7(cosO  -f  isinO). 
(i)    2  V3(cos  I  w  + 1  sin  I  tt). 
(k)  ll(cos  ^  TT  +  I  sin  I  w). 


(6)  5(cos  I  IT  -\- i  sin  ^ -it)  . 
(d)  4(cos  I TT  +  I  sin  I  tt). 
(/)   V3(cos  f  TT  +  I  sin  I  tt) . 
(h)  5(cos7r  +  isin7r). 
(j)  5  V2  (cos  I TT  +  1  sin  I  tt)  . 
(0    8(cos75°  +  isin75°). 


3.  Put  the  following  complex  numbers  in  polar  form,  perform  the 
indicated  multiplication  or  division,  and  write  the  result  in  the  form 
X  +  yi'     Check  by  algebra  and  illustrate  by  geometry. 

(a)   (2V3+2  0(3  4-3V3  0.  (^)    (1  +  0(2  +  20- 


(c)  (-2-20(5  +  50- 
(e)  (1+V30(1-V30 
2VS-2i. 


U) 


bi 


1+i 
\-i 


(h) 
(k) 


4i 


(d)   (_4  +  4V3  0(-3-3\/3  0- 
(/)    (-2)(-3  0. 

-7 


5+  5i 

1 
-  \/3  -  i 


(O 


3  +  3V3  4 


(0  —■ 


126  PLANE  ANALYTIC  GEOMETRY       [VII,  §  125 

4.  Show  that  the  modulus  of  the  product  of  the  complex  numbers 
a  +  hi  and  c  +  di  is  y/(^a^  +  b'^)(^c^  +  d^). 

6.  Show  by  geometric  construction  that  the  product  of  two  conjugate 
complex  numbers  is  a  real  number. 

6.  Show  how  to  locate  by  geometric  construction  the  point  which 
represents  the  reciprocal  of  a  complex  number. 

7.  Show  that  the  point  P  that  represents  a  complex  number  z  and 
the  point  P'  that  represents  the  conjugate  of  the  reciprocal  \/z  are  inverse 
points  with  respect  to  the  unit  circle  about  the  origin. 

8.  With  respect  to  the  unit  circle  about  the  origin,  find  the  complex 
numbers  representing  the  points  inverse  to 

(a)  3  +  4i.         (6)  3+V^^.         (c)   -  5  +  3  i.         (d)  1  -  6  i. 

9.  Show  that  the  ratio  of  two  complex  numbers  whose  amplitudes 
differ  by  ±  |  tt  is  an  imaginary  number. 

10.  Show  that  the  ratio  of  two  complex  numbers  whose  amplitudes 
are  equal  or  differ  by  ±  t  is  a  real  number. 

126.  De  Moivre'S  Theorem.  The  rule  for  multiplying  two  com- 
plex numbers  (§  124)  gives  at  once  for  the  square  of  a  complex  number 
z  =  r(cos  <f>  +  i  sin  0) : 

z^  =  [r(cos0  +  isin0)]2  =  r^(^cos2  0  +  isin2  0). 

Multiplying  both   members   by  ^  =  r(cos  0  +  i  sin  0)    we  find  for  the 

cube : 

z^  =  [r(cos  0  +  1  sin  0)]^  =  r3(cos  3  0  +  i  sin  3  0) . 

This  suggests  that  we  have  generally  for  the  ?ith  power  of  0,  n  being 
any  positive  integer : 

zn  —\r  (cos  0  +  1  sin  0)]"  =  r'»(cos  n  0  +  i  sin  n  0). 
This  is  known  as  de  Moivre' s  formula. 

To  complete  the  formal  proof  we  use  mathematical  induction  (§  62) . 
Assuming  the  formula  to  hold  for  some  particular  value  of  w,  it  is  at  once 
shown  to  hold  for  w+  1,  by  multiplying  both  members  by 

z  =  »'(cos0  +  isin0) 
which  gives 

0«+i=[r(cos0  +  I  sin  0)]"+^  =  r"+i[cos  (w  +  l)0  +  isin  (w  +  l)0]. 
As  the  formula  holds  for  w  =  2,  it  holds  for  n  =  3,  and  hence  for  w  =  4,  etc., 
i.e.  for  any  positive  integer. 


VII,  §  128]  COMPLEX  NUMBERS  127 

127.  Generalization  of  De  Moivre's  Theorem.  De  Moivre's 
formula  can  be  shown  to  hold  for  any  real  exponent  n.  That  it  holds  for 
a  negative  integer  is  seen  as  follows  : 

If  in  the  formula  for  the  quotient  z  =  Zi/z2  (§  126)  we  put  ri  =  1, 
01  =  0,  we  find 

—  =  —  (cos  02  —  i  sin  02), 

or  dropping  the  subscript  2  : 

-  =  -  (cos  <t>  —  i  sin  0) , 
z      r 

If  we  raise  this  complex  number  to  the  nth  power  {n  being  a  positive 
integer) ,  which  can  be  done  by  §  126,  we  find 


(i)' 


z-^  =  — (cos«0  —  I  sin  w0), 


which  proves  de  Moivre's  formula  for  a  negative  integral  exponent. 
If  in  de  Moivre's  formula  (§  126)  we  put 

a 

ntjy  =  d,  1"^  =  p,  and  hence  0  =  -,  r  =  v^, 

n 

where  y/p  is  the  positive  nth  root  of  the  real  number  p,  we  obtain 
I  \//)f  cos-  +  /sin- I  I   =/)(cos^  +  isin  0), 

i.e.  [/)(cos  d  +  i  sin  6)^=  Vplcos  -  +  i  sin-V 

\       n  n) 

This  shows  that  de  Moivre's  formula  holds  when  the  exponent  is  of  the 
form  \/n.  The  extension  to  the  case  when  the  exponent  is  any  rational 
fraction  is  then  obvious. 

128.  Imaginary  Roots.  The  last  formula  gives  a  means  of  finding 
an  will  root  of  any  real  or  complex  number.  To  find  all  the  roots  of  a 
complex  number  z  =  p(cos  6  -\-  i  sin  d)  we  must  observe  that  as 

cos  d  =  cos  (6  +  2  TTw),  sin  d  =  sin  (^  +  2  irm), 

where  m  is  any  integer,  the  number  z  can  be  written  in  the  form 

z  =  p[cos  (^  -f  2  7rm)  +  /  sin  (^  +  2  Trm)], 

so  that  by  §  127  its  roots  are  given  by 


128 


PLANE  ANALYTIC  GEOMETRY       [VII,  §  128 


cos  ^±-2^+ I  sin  ^±1^ 


If  in  this  expression  we  give  to  m  successively  all  integral  values,  it  takes 
just  n  different  values,  viz.  those  for  7>i  =  0, 1,  2,  •••  ,  n  —  1 ;  therefore  any 
complex  number  z  =  p(cos  6  -\-  i  sin  6)  lias  n  roots,  viz. : 


Vpfcos^  +  isin^V  ^pfcos^^t^+isini±^), 
\       n  111         V  '*  *i      ' 


..  Vp[i 


+  (n-l)2 


+  i  sin 


g  +  (H-l)27r' 


These  n  roots  all  have  the  same  modulus  \/p,  while  the  amplitudes  differ 

by  2  ir/n.     Hence  the  points  representing  these  n 

roots   lie  on  a  circle  of  radius  Vp  about  the  origin 

and  divide  this  circle  into  n  equal  parts. 

For  example,  the  three  cube  roots  of  8  i  are  found 

as  follows.    In  polar  form 

f 

0  +  8  I  =  8(cos  ^  TT  +  i  sin  I  tt)  ; 

by  de  Moivre's  formula  (§  127)  we  have 
[8(cos  ^  TT  +  z  sin  ^  tt)] 3 
=  2[cos  iZ_i_2=  +  i  sin  i^^Jl^jmJ, 

=  2[cos  (i  TT  +  f  irm)  +  i  sin  (^  tt  +  |  Trm)]  ; 
w  =  0  gives  the  root : 

w  =  1  gives  the  root :  2(cos  f  tt  +  i  sin  f  tt)  =  2(  —  ^\/3+i  \)  =  —  VS  +  i ; 
w  =  2  gives  the  root :  2(cos  |  tt  +  i sin  |  tt)  =  2(0  +  i  (  -  1))  =  —  2  i. 

If  we  put  w  =  3,  we  get  the  first  root  again,  wi  =  4  gives  the  second  root, 
and  so  on.  Thus  there  are  three  distinct  cube  roots  of  8  i,  viz.  VS  +  i, 
\/3  + 1,  —  2  i.  These  roots  are  represented  by  the  points  Pi,  P2,  Pa* 
respectively  (Fig.  51). 


Fig.  61 


VII,  §129]  COMPLEX  .NUMBERS  129 

129.  Square  Roots.  The  particular  problem  of  finding  the  square 
root  of  a  complex  number  a  +  &i  can  also  be  solved  by  observing  that  the 
problem  requires  us  to  find  a  complex  number  x  +  yi  such  that 

a  +  6i  =  (x  +  yiy. 
Expanding  the  square  and  equating  real  and  imaginary  parts,  we  find  for 
the  determination  of  x  and  y  the  two  equations 

x^-y^=a,         2  xy  =  h. 
Eliminating  y  between  these  two  equations,  we  obtain 

a;2  -  —  =a ;   that  is,  a;*  -  ax2  -  i  fo2  _  q  • 


whence  Xi^  =  |(a  +  Va^  +  h-),        x-^  =  ^(a  -  Va^  +  62). 

Since  x  is  to  be  a  real  number  and  hence  x^  must  be  positive,  and  as 
a<A/a2+62  (unless  6  =  0,  which  ^ould  mean  that  the  given  number  a  +  bi 
is  real),  we  must  take  Xi2  and  not  X2^.     Hence 


x=±  v|(a+Va2+62). 

These  values  of  x  are  zero  only  when  b=0  and  a  <  0  ;  for  then  Va^  =  —  a. 
In  this  particular  case  we  find  y  =  ±V—  a,  and  hence 

Va  +  bi  =±  V—  a  i. 
In  the  general  case,  when  &  ^t  0,  we  find  from  the  equation  2xy  =  b  for 
each  of  the  two  values  of  x  one  value  of  y. 

'  EXERCISES 

1.  Show  how  to  locate  the  square  of  a  complex  number  by  geometric 
construction.     Locate  the  cube. 

2.  Show  geometrically  that  8i  (Fig.  51)  is  the  product  of  the  numbers 
represented  by  the  points  Pi,  P2,  P3. 

3.  For  zi  =  l  -\-2i,  Z2—-2  —  i  show  that  z{^  —  z^^  ={z\  +  2^2) (2^1— 02) 
and  illustrate  geometrically. 

4.  For  the  same   numbers  verify  and   illustrate  geometrically  that 

{Z\  —  ZiY  -Z^^2  ZxZi  +  02^. 

6.  Show  how  to  locate  the  points  that  represent  the  square  roots  of  a 
complex  number. 

6.  Locate  by  geometric  construction  in  two  "ways  the  points  which 
represent  [r(cos 0  +  i sin  0)]^. 

K 


130  PLANE  ANALYTIC  GEOMETRY      [VII,  §  129 

7.  Put  the  following  complex  numbers  in  polar  form,  perform  the  in- 
dicated operations,  and  check  by  geometric  construction : 

(a)    (1  +  V3i)2.  (b)  (-1  +  0^.  (c)    (-V3-i)2. 

id)    (V3  +  i)^.  (e)  (-  1)^.  (/)  (-0^. 

(g)    Vl+V3i.  (/i)  -yZ-l-y/Si.  (i)    ^-2-\-2VSi. 

U)     ^/-'^-Su  (A)  v/-4-h4i.  (Z)    ^/6il. 

(m)  V'^Hol.  (w)  \/87.  (o)   V(-3i)». 

8.  Find  the  square  roots  of  each  of  the  following  complex  numbers 
by  using  the  method  of  §  129 : 

(a)  7  +  24  i.  (6)  4 1.  (c)    -2(8  +  151). 

(d)  -  16.  (e)  j\(5  -  12  i).  (/)  4  a6  +  2(a2  -  &2)i-. 
(gf)  _  2[2  a&  +  (a2  -  62) ^-j.                                (^)  _  4  ^2^,2  4.  2(0*  -  6*)  i. 

9.  Find  the  three  cube  roots  of  unity  and  show  that  either  complex 
root  is  the  square  of  the  other,  i.e.  if  one  complex  root  of  unity  is  denoted 
by  w,  the  other  is  w2.     The  three  cube  roots  of  unity  then  are  1,  w,  u^. 

10.  If  1 ,  w,  w2  are  the  cube  roots  of  unity  (see  Ex.  9)  show  that : 

(a)    1  =  w^  =  w^  =  w^",  n  being  an  integer. 
(6)    l  +  w  +  a>2  =  0. 

(C)      (1  +  w2)4  =  W. 

(d)  (a;i>  +  a;2g)  (0,2^  4.  ^^g)  (p^  g)  ^  p8  +  ^3. 

(e)  (1  _a;  +  w2)(l  +  w-a,2)  ^4. 

(/)    (1  _  w  +  a;2)  (1  -  0,2  4-  0,4)  (1  _  0,4  +  w8)  =  _  8  0,. 

11.  Prove  de  Moivre's  formula  for  n  any  rational  fraction,  i.e.  show 
that,  if  p,  g,  w,  are  integers, 

[r(cos  0  +  i sin  0)]^  = /.«  fcos^^Jl^^  +  I  sin£^±l^l 
L  g  g        J 

12.  Show  by  geometric  construction  that  the  sum  of  the  three  cube 
roots  of  any  number  is  equal  to  zero  ;  that  the  sum  of  the  four  fourth 
roots  is  zero. 

13.  Solve  the  following  equations  and  locate  the  points  which  repre- 
sent the  roots : 

(a)a:2-l=0.    •  (6)   x^  +  1  =  0.       (c)  x*  -  1  =  0.       (d)x^-l=0. 

(e)  a:«  -  1  =  0.       (  f)  x^  -  27  =  0.      (g)  x^  +  1=0.       (/i)  x*  +  16  =  0. 
(i)   x5  +  32  =  0.     (j)    x2  +  a2  =  0.      (A;)  x^  +  ^3  =  0.      (0   x^  -  1  =  0. 


CHAPTER   VIII 

POLYNOMIALS.      NUMERICAL  EQUATIONS 

PART   I.     QUADRATIC   FUNCTION  —  PARABOLA 

130.  Linear  Function.  As  mentioned  in  §  28,  an  expression 
of  the  form  mx  +  h,  where  m  and  h  are  given  real  numbers 
(m=fzO)  while  ic  may  take  any  real  value,  is  called  a  linear 
function  of  x.  We  have  seen  that  this  function  is  represented 
graphically  by  the  ordinates  of  the  straight  line 

y  =  mx  4-  b ; 

b  is  the  value  of  y  for  x  =  0,  and  m  is  the  slope  of  the  line,  i.e. 
the  rate  of  change  of  the  function  y  with  respect  to  x. 

131.  Quadratic  Function.  Parabola.  An  expression  of 
the  form  aa^  -\-bx  +  c  in  which  a  ^  0  is  called  a  quadratic  func- 
tion of  X,  and  the  curve 

y  =  ax"^  -\-bx-{-Cj 

whose  ordinates  represent  the  function,  is  called  a  parabola. 

If  the  coefficients  a,  b,  c  are  given  numerically,  any  number 
of  points  of  this  curve  can  be  located  by  arbitrarily  assigning 
to  the  abscissa  x  any  series  of  values  and  computing  from  the 
equation  the  corresponding  values  of  the  ordinates.  This 
process  is  known  as  plotting  the  curve  by  points  ;  it  is  some- 
what laborious;  but  a  study  of  the  nature  of  the  quadratic 
function  will  show  that  the  determination  of  a  few  points  is 
sufficient  to  give  a  good  idea  of  the  curve. 

131 


132 


PLANE  ANALYTIC  GEOMETRY      [VIII,  §  132 


Fig.  52 


132.  The  Form  y  =  ax".     Let  us  first  take  6  =  0,  c  =  0 ;  the 
resulting  equation 

(1)  y  =  ax"^ 

represents  a  parabola  which  passes  through  the  origin,  since 
the  values  0,  0  satisfy  the  equation.  This  x>ardbola  is  symmet- 
ric ivith  respect  to  the  axis  Oy  ;  for,  the  values  of  y  correspond- 
ing to  any  two  equal  and  opposite  values  of  x  are  equal.  This 
line  oi  symmetry  is  called  the  axis  of  the 
parabola ;  its  intersection  with  the  parab- 
ola is  called  the  vertex. 

We  may  distinguish  two  cases  accord- 
ing as  a  >  0  or  a  <  0 ;  if  a  =  0,  the  equa- 
tion becomes  2/  =  0,  which  represents  the 
axis  Ox. 

(1)  If  a  >  0,  the  curve  lies  above  the  axis  Ox.  For,  no  matter 
what  positive  or  negative  value  is  assigned  to  x,  y  is  positive. 
Furthermore,  as  x  is  allowed  to  increase  in  absolute  value,  y 
also  increases  indefinitely.  Hence  the  parabola  lies  in  the  first 
and  second  quadrants  with  its  vertex  at 
the  origin  and  opens  upward,  i.e.  is  con- 
cave upward  (Fig.  52). 

(2)  If  a<0,  we  conclude,  similarly, 
that  the  parabola  lies  below  the  axis  Ox, 
in  the  third  and  fourth  quadrants,  with 
its  vertex  at  the  origin  and  opens  down- 
ward, i.e.  is  concave  downward  (Fig.  53). 

Draw  the  following  parabolas: 

y  =  x',y  =  ^x',y^-^o?,y^\x'. 

133.  The  General  Equation.     The  curve  represented  by  the 
more  general  equation 

(2)  y  =  ojx?  +  hx  •{- c 

differs  from  the  parabola  y  —  a^?  only  in  position.    To  see  this 


Fig.  53 


VIII,  §134]     POLYNOMIALS  — THE  PARABOLA 


133 


we  use  the  process   of   completing  the  square  in  x\   i.e.  we 
write  the  equation  in  the  equivalent  form 


y 


I.e. 


y- 

If  we  put 


7.  +  ^)  =  K''^^)- 


4 


^' 


2  a  4  a 


Fig.  54 


the  equation  becomes 

y  —  k  =  a(x  —  hy, 
and  it  is  clear  (§  13)  that,  with  reference  to  parallel  axes 
OiXi,  Oi2/i  through  the  point  Oi  (Ji,  k)  the  equation  of  the 
curve  is  y-^  =  ax^  (Fig.  54).  The  parabola  (2)  has  therefore 
the  same  shape  as  the  parabola  y  =  ax"^ ;  but  its  vertex  lies  at 
the  point  {h,  k),  and  its  axis  is  the  line  x  =  h.  The  curve 
opens  upward  or  downward  according  as  a  >  0  or  a  <  0. 

134.  Nature  of  the  Curve.  To  sketch  the  parabola  (2) 
roughly,  it  is  often  sufficient  to  find  the  vertex  (by  completing 
the  square  in  x,  as  in  §  133),  and  the  intersections  with  the  axes. 
The  intercept  on  the  axis  Oy  is  obviously  equal  to  c.  The  in- 
tercepts on  the  axis  Ox  are  found  by  solving   the   quadratic 

equation 

ax"^  -{-  bx  -j-  c  =  0. 

We  have  thus  an  interesting  interpretation  of  the  roots  of  any 
quadratic  equation  :  the  roots  of  ax^  -f-  6rc  +  c  =  0  are  the 
abscissas  of  the  points  at  which  the  parabola  (2)  intersects 
the  axis  Ox.  The  ordinate  of  the  vertex  of  the  parabola 
is  evidently  the  least  or  greatest  value  of  the  function 
y  =  ax^  -\-hx-\-c  according  as  a  is  greater  or  less  than  zero. 


134  PLANE  ANALYTIC  GEOMETRY     [VIII,  §  134 

EXERCISES 

1.  With  respect  to  the  same  coordinate  axes  draw  the  curves  y  =  ax^ 
for  a=2^  f,  1,  I,  0,  —  ^,  —  1,  —  |,  —  2.  What  happens  to  the  parabola 
y  =  ax^  as  a  changes  ? 

2.  Determine  in  each  of  tlie  following  examples  the  value  of  a  so  that 
the  parabola  y  =  ax^  will  pass  through  the  given  point : 

(a)   (2,3).  (6)   (-4,1).  (c)   (-2,  -2).  (d)   (3,-4). 

3.  A  body  thrown  vertically  upward  in  a  vacuum  with  a  velocity  of  v 
feet  per  second  will  just  reach  a  height  of  h  feet  such  that  h  =  ^^^  v^. 
Draw  the  curve  whose  ordinates  represent  the  height  as  a  function  of  the 
initial  velocity. 

(a)  With  what  velocity  must  a  ball  be  thrown  vertically  upward  to  rise 
to  a  height  of  100  ft.  ? 

(6)  How  high  will  a  bullet  rise  if  shot  vertically  upward  with  an  ini- 
tial velocity  of  800  ft.  per  sec. ,  the  resistance  of  the  air  being  neglected  ? 

4.  The  period  of  a  pendulum  of  length  I  {i.e.  the  time  of  a  small 
back  and  forth  swing)  is  r=  2iry/l/g.  Take  g  =  S2  ft. /sec.  and  draw 
the  curve  whose  ordinates  represent  the  length  I  of  the  pendulum  as  a 
function  of  the  period  T. 

(a)  How  long  is  a  pendulum  that  beats  seconds  (i.e.  of  period  2  sec.)  ? 
(6)  How  long  is  a  pendulum  that  makes  one  swing  in  two  seconds  ? 
(c)  Find  the  period  of  a  pendulum  of  length  one  yard. 

5.  Draw  the  following  parabolas  and  find  their  vertices  and  axes : 
(a)  y  =  lx^-x  +  2.         (h)  y  =  -lx^  +  x.        (c)  y  =  5x^  +  lbx  +  3. 
(d)  y  =  2-x-x^  (e)   2/  =  a;2  -  9.  (f)y  =  -9-  x\ 

(^)  y=3a;2_6«  +  5.      (/i)  y  =  |a;2  +  2a;  -  6.         {i)  x'^  -  2x -y  =  () . 

6.  What  is  the  value  of  h  if  the  parabola  y  =  x^  -\-  bx  —  6  passes 
through  the  point  (1,  5)  ?  of  c  if  the  parabola  y  =  x!:^  --Qx -\-  c  passes 
through  the  same  point  ? 

7.  Suppose  the  parabola  y  =  ax^  drawn ;  how  would  you  draw  y  = 
a  (x+2)2  ?  y  =  a(x-7)2  ?  y  =  ax2  +  2  ?  y  =  a.r2  -  7  ?  y  =  ax2+  2x4-3? 

8.  What  happens  to  the  parabola  y  =  ax'^  +  hx  +  c  as  c  changes  ? 
For  example,  take  the  parabola  y  =  x2  —  x  +  c,  where  c  =  —  3,  —  2,  —  1, 
0,  1,  2,  3. 


VIII,  §134]     POLYNOMIALS  — THE  PARABOLA  135 

9.  What  happens  to  the  parabola  y  =  ax-  +  bx  -\-  c  as  a  changes  ? 
For  example,  take  y  =  ax'^  —  x  —  6,  where  «  =  2,  1,  |,  0,  —  ^,  —  1,  —  2. 

10.  (a)  If  the  parabola  y  =  ax^  +  bx  is  to  pass  through  the  points 
(1,  4),  (—  2,  1)  what  must  be  the  values  of  a  and  b  ?  (6)  Determine  the 
parabola  y  =  ax^  +  bx  +  c  so  as  to  pass  through  the  points  (1,  ^),  (3,  2), 
(4,  f )  ;  sketch. 

11.  The  path  of  a  projectile  in  a  vacuum  is  a  parabola  with  vertical 
axis,  opening  downward.  With  the  starting  point  of  the  projectile  as 
origin  and  the  axis  Ox  horizontal,  the  equation  of  the  path  must  be  of  the 
form  y  =  ax^  +  bx.  If  the  projectile  is  observed  to  pass  through  the  points 
(30,  20)  and  (50,  30),  what  is  the  equation  of  the  path?  What  is  the 
highest  point  reached  ?    Where  will  the  projectile  reach  the  ground  ? 

12.  Find  the  equations  of  the  parabolas  determined  by  the  following 
conditions : 

(a)  the  axis  coincides  with  Oy,  the  vertex  is  at  the  origin,  and  the 
curve  passes  through  the  point  (—2,  —  3)  ; 

(6)  the  axis  is  the  line  x  =  3,  the  vertex  is  at  (3,  —  2),  and  the  curve 
passes  through  the  origin  ; 

(c)  the  axis  is  the  line  aj  =—  4,  the  vertex  is  (—  4,  6),  and  the  curve 
passes  through  the  point  (1,  2). 

13.  Sketch  the  following  parabolas  and  lines  and  find  the  coordinates 
of  their  points  of  intersection  : 

(a)  y  =  6x%y  =  'Jx-\-S.  (^b)  y  =  2  x^  -  3x,  y  =  x  -\-  6.   » 

(c)  y  =  2-3x^,y  =  2x-\-S.  (d)  y  =  S -\- x- x^,  x  +  y  -  4  =  0. 

14.  Sketch  the  following  curves  and  find  their  intersections : 

(a)  x2  +  y2  =  25,  y  =  |x2.  (&)  x:^-\-y2-6y  =  0,y  =  ^x^-2x  +  e. 

15.  The  ordinate  of  every  point  of  the  line  y  :^  |  a;  +  4  is  the  sum  of 
the  corresponding  ordinates  of  the  lines  y  =  ^x  and  y  =  4.  Draw  the  last 
two  lines  and  from  them  construct  the  first  line. 

16.  The  ordinate  of  every  point  of  the  parabola  y  =  lx^  +  ^x— 1  is 
the  sum  of  the  corresponding  ordinates  of  the  parabola  y  =  ^x^  and  the 
line  y  =  ^x  —  l.    From  this  fact  draw  the  former  parabola. 

17.  The  ordinate  of  every  point  of  the  parabola  y  =  ^x^  —  x  +  Sis  the 
difference  of  the  corresponding  ordinates  of  the  parabola  y  =  ^x^  and  the 
line  2/  =  X  —  3,     In  this  way  sketch  the  former  parabola. 


136  PLANE  ANALYTIC   GEOMETRY     [VIII,  §  134 

18.  Suppose  the  parabola  y  =  ax^  +  bx  -^  c  drawn,  how  would  you 
sketch  the  following  curves  ?    Are  these  curves  also  parabolas  ? 

(a)  y  =  a(x-\-  hY  +  h{x  +  h)+c,h> 0. 

(&)  y  =  a(x-  2)2  +  5(x  -  2)  +  c. 

(c)  2/  =  a(2x)2  +  6(2a;)4-c. 

(d)  y  =  a(^\xy  +  b(i\x)+c. 

19.  Find  the  values  of  x  for  which  the  following  relations  are  true  : 
(a)  a:2  _  a;  -  12  <  0.  (6)    12-a;-a:2>0. 

(c)   3x2  +  6a;-2^0.  {d)    5  +  13x-6x2^0. 

(e)  «2_5>3a;  +  6.  (/)  x2-5<3x  +  5. 

20.  Show  that  the  equation  of  the  parabola  y  =  ax'^  -\- hx -{■  c  that 

passes  through  the  points  {x\ ,  yi),  (x^ ,  ^2),  (a^3 ,  yi)  may  be  written  in 

the  form 

y     x^     X      \ 


yi 

:«i2 

a:i     1 

2/2 

3^2^ 

X2      1 

2/3 

X32 

X3      1 

(a)  Show  that  if  the  minor  of  x'^  vanishes,  the  three  given  points  lie  on 
a  line. 

(6)  What  conclusion  do  you  draw  if  the  minor  of  y  vanishes  ? 

(c)  To  what  does  this  determinant  reduce  if  the  origin  is  one  of  the 
given  points  ? 

135.  Sjonmetry.  Two  points  P^ ,  P^  are  said  to  be  situated 
symmetrically  with  respect  to  a  line  Z,  if  I  is  the  perpendicular 
bisector  of  P^P^ ;  this  is  also  expressed  by  saying  that  either 
point  is  the  reflection  of  the  other  in  the  line  I. 

Any  two  plane  figures  are  said  to  be  symmetric  with  respect 
to  a  line  I  in  their  plane  if  either  figure  is  formed  of  the  reflec- 
tions in  I  of  all  the  points  of  the  other  figure.  Each  figure  is 
then  the  reflection  of  the  other  in  the  line  I.  Two  such  figures 
are  evidently  brought  to  coincidence  by  turning  either  figure 
about  the  line  I  through  two  right  angles.  Thus,  the  lines 
2/  =  2 ic -h 3  and  y  =  —  2x  —  S  are  symmetric  with  respect  to 
the  axis  Ox. 


VIII,  §135]     POLYNOMIALS— THE  PARABOLA  137 

A  line  /  is  called  an  axis  of  symmetry  (or  simply  an  axis)  of 
a  figure  if  the  portion  of  the  figure  on  one  side  of  I  is  the 
reflection  in  I  of  the  portion  on  the  other  side.  Thus,  any 
diameter  of  a  circle  is  an  axis  of  symmetry  of  the  circle. 
What  are  the  axes  of  symmetry  of  a  square  ?  of  a  rectangle  ? 
of  a  parallelogram  ? 

In  analytic  geometry,  symmetry  with  respect  to  the  axes  of 
coordinates,  and  to  the  lines  y=±x,isoi  particular  importance. 

It  is  readily  seen  that  if  a  figure  is  symmetric  with  respect 
to  both  axes  of  coordinates,  it  is  symmetric  with  respect  to  the 
origin^  i.e.  to  every  point  Pi  of  the  figure  there  exists  another 
point  Pg  of  the  figure  such  that  the  origin  bisects  PiP^.  A 
point  of  symmetry  of  a  figure  is  also  called  center  of  the  figure. 

EXERCISES 

1.  Give  the  coordinates  of  the  reflection  of  the  point  (a,  &)  in  the 
axis  Ox ;  in  the  axis  Oy ;  in  the  line  y  =  X]  in  the  line  y  =  2  x ;  in  the 
line  y  =—  X. 

2.  Show  that  when  x  is  replaced  by  —  x  in  the  equation  of  a  given 
curve,  we  obtain  the  equation  of  the  reflection  of  the  given  curve  in  the 
y-axis. 

3.  Show  that  when  x  and  y  are  replaced  by  y  and  x,  respectively,  in  the 
equation  of  a  given  curve,  we  obtain  the  equation  of  the  reflection  of  the 
given  curve  in  the  line  y  =  x. 

4.  Sketch  the  lines  y  =  —  2x  +  6  and  x  =  —  2.y  -\-  5  and  find  their 
point  of  intersection. 

6.  Sketch  the  parabolas  y  =  x^  and  x=  y^  and  find  their  points  of  in- 
tersection. 

6.  Find  the  equation  of  the  reflection  of  the  line  2x  —  3y-|-4  =  0in 
the  line  y  =  x;  in  the  axis  Ox;  in  the  axis  Oy  ;  in  the  line  y  —  —x. 

7.  What  is  the  reflection  of  the  line  x  =  a  in  the  line  y  =  x?  in  the 
axes? 

8.  Find  and  sketch  the  circle  which  is  the  reflection  of  the  circle 
x2  -I-  y2  _  3  ^  _  2  =  0  in  the  line  y  =  x,  and  find  the  points  in  which  the 
two  circlea  intersect. 


138 


PLANE  ANALYTIC  GEOMETRY     [VIII,  §  135 


9.  Find  the  circle  which  is  the  reflection  of  the  circle  x^  -\-y'^  —ix  +3 
=  0  in  the  line  y  =  x;  in  the  coordinate  axes.  Sketch  all  of  these 
circles. 

10.  What  is  the  general  equation  of  a  circle  which  is  its  own  reflection 
in  the  line  y  =  x?  in  the  axis  Ox  ?  in  the  axis  Oy  '?  What  circle  is  its 
own  reflection  in  all  three  of  these  lines  ? 

11.  What  is  the  equation  of  the  reflection  of  the  parabola  y  =—x^  +  4: 
in  the  line  y  =  x?  in  the  line  y  =  —  x?     Are  these  reflections  parabolas  ? 

12.  What  is  the  reflection  of  the  parabola  ?/  =  3  ic'-^  —  5  x  +  6  in  the  axis 
Ox  ?  in  the  axis  Oy  ?    Are  these  reflections  parabolas  ? 

13.  By  drawing  accurately  the  parabolas  y  -\- x^  =  1,  x -{- y^  =  11,  find 
approximately  the  coordinates  of  their  points  of  intersection. 

14.  If  the  Cartesian  equation  of  a  curve  is  not  changed  when  x  is  re- 
placed by  —  X,  the  curve  is  symmetric  with  respect  to  Oy  ;  if  it  is  not 
changed  when  y  is  replaced  by  —  y,  the  curve  is  symmetric  with  respect 
to  Ox  ;  if  it  is  not  changed  when  x  and  y  are  replaced  by  —  x  and  —  y, 
respectively,  the  curve  is  symmetric  with  respect  to  the  origin  ;  if  it  is 
not  changed  when  x  and  y  are  interchanged,  the  curve  is  symmetric  with 
respect  to  y  =  x. 

136.  Slope  of  Secant.  Let  P(a;,  y)  be  any  point  of  the 
parabola 

(1)  y  =  ax\ 
If  Pi(xi ,  2/i)  be  any  other  point  of 
this  parabola  so  that 

(2)  2/1  =  ctx,^ 
the  line  PPi  (Fig.  55)  is  called  a 
secant. 

For  the  slope  tan  Oj  of  this  secant 
we  have  from  Fig.  55 : 

(3) 

or,  substituting  for  y  and  i/i  their  values  : 

(4)  tan  «i  =  ^W  -  ^')  ^  a{x  +  x^) 

Xy—  X  ^r— : r 


SQi      x^  —  X        Aa; 


Vm,  §138]    POLYNOMIALS  — THE  PARABOLA  139 

137.  Slope  of  Tangent.  Keeping  the  point  P  (Fig.  6b) 
fixed,  let  the  point  Pi  move  along  the  parabola  toward  P;  the 
limiting  position  which  the  secant  PP^  assumes  at  the  instant 
when  Pi  passes  through.  P  is  called  the  tangent  to  the  parabola 
at  the  point  P. 

Let  us  determine  the  slope  tana  of  this  tangent.  As  the 
secant  turns  about  P  approaching  the  tangent,  the  point  Qi  ap- 
proaches the  point  Q,  and  in  the  limit  OQi  =  Xi  becomes  OQ=x. 
The  last  formula  of  §  136  gives  therefore  tan  a  if  we  make 

Xi  =  x: 

tan  a  =  2 


The  slope  of  the  tangent  at  P  which  indicates  the  '•  steep- 
ness "  of  the  curve  at  P  is  also  called  the  slojje  of  the  parabola 
at  P.  Thus  the  slope  of  the  parabola  y  =  ax^  at  any  point 
whose  abscissa  is  a;  is  =2 ax-,  notice  that  it  varies  from  point 
to  point,  being  a  function  of  x,  while  the  slope  of  a  straight 
line  is  constant  all  along  the  line. 

The  knowledge  of  the  slope  of  a  curve  is  of  great  assistance 
in  sketching  the  curve  because  it  enables  us,  after  locating 
a  number  of  points,  to  draw  the  tangent  at  each  point.  Thus, 
for  the  parabola  ?/  =  |  aj^  we  find  tan  a  =  ^x  ;  locate  the  points 
for  which  a?  =  0, 1,  2,  —  1,  —  2,  and  draw  the  tangents  at  these 
points ;  then  sketch  in  the  curve. 

138.  Derivative.  If  we  think  of  the  ordinate  of  the  parab- 
ola y  =  ax^  as  representing  the  function  ax^,  the  slope  of  the 
parabola  represents  the  rate  at  which  the  function  varies  with 
X  and  is  called  the  derivative  of  the  function  ax"^.  We  shall 
denote  the  derivative  of  y  by  y'.  In  §  137  we  have  proved 
that  the  derivative  of  the  function 

y  =  ax^, 
is  y'  =  2  ax. 


140  PLANE  ANALYTIC  GEOMETRY     [VIII,  §  138 

The  process  of  finding  the  derivative  of  a  function,  which  is 
called  differentiation,  consists,  according  to  §§  136-137,  in  the 
following  steps :  Starting  with  the  value  y=  ax^  of  the  func- 
tion for  some  particular  value  of  x  (say,  at  the  point  P,  Fig.  55), 
we  give  to  x  an  increment  x^—x  =  ^x  (compare  § 9)  and 
calculate  the  value  of  the  corresponding  increment  y^—y^Ay 
of  the  function.  Then  the  derivative  ?/'  of  the  function  y  is  the 
limit  that  Ay  /  Ax  approaches  as  Ax  approaches  zero.  In  the 
case  of  the  function  y  =  aa^  we  have 

Ay=y^-y  =  a{x^^  -  x"^)  =  a[(x  +  Axy  -  a;^]  =  a[2  xAx  +  (Axy^ ; 

hence  —  =  a(2  x  +  Ax). 

Ax        ^       ^       ^ 

The  limit  of  the  right-hand  member  as  Ax  approaches  zero 

gives  the  derivative : 

y'  =  2ax.- 

Thus,  the  area  y  of  a  circle  in  terms  of  its  radius  x  is 

y  =  irx^. 
Hence  the  derivative  y',  that  is  the  slope  of  the  tangent  to  the  curve  that 
represents  the  equation  y  =  ttx^,  is 

y'  =2  irx. 
This'represents  (§  137)  the  rate  of  increase  of  the  area  y  with  respect  to  x. 
Since  2  ira;  is  the  length  of  the  circumference,  we  see  that  the  rate  of  in- 
crease of  the  area  y  with  respect  to  the  radius  x  is  equal  to  the  circumfer- 
ence of  the  circle. 

139.  Derivative  of  General  Quadratic  Function.  By  this 
process  we  can  at  once  find  the  derivative  of  the  general  quad- 
ratic function  y  =  aa^ -{- bx  -\-  c  (§  131),  and  hence  the  slope  of 
the  parabola  represented  by  this  equation.     We  have  here 

Ay  =  a(x  +  Axy -f-  b(x  -f-  Ax)  -{-c  —  {ax^ -\-hx-\-c) 
=  2  ax  Ax  -\-  a{Axy  -\-  bAx ; 

hence       —  =  2  ax-{-b  -\- aAx. 
Ax 


VIII,  §140]     POLYNOMIALS  — THE  PARABOLA  141 

The  limit,  as  Ax  becomes  zero,  is  2ax-\-  b;  hence  the  deriva- 
tive of  the  quadratic  function  y  =.ax^  -\-hx-\-  cis 

y^  =2ax  +  h. 

140.  Maximum  or  Minimum  Value.  It  follows  both  from 
the  definition  of  the  derivative  as  the  limit  of  Ap/Ax  and  from 
its  geometrical  interpretation  as  the  slope,  tana,  of  the  curve 
that  if,  for  any  value  of  x,  the  derivative  is  positive,  the  function, 
i.e.  the  ordinate  of  the  curve,  is  {algebraically)  increasing;  if 
the  derivative  is  negative,  the  function  is  decreasing. 

At  a  point  where  the  derivative  is  zero  the  tangent  to  the  curve 
is  parallel  to  the  axis  Ox.  The  abscissas  of  the  points  at  which 
the  tangent  is  parallel  to  Ox  can  therefore  be  found  by  equat- 
ing the  derivative  to  zero.  In  this  way  we  find  that  the 
abscissa  of  the  vertex  of  the  parabola  y  =  ax^  -|-  6a;  -f  c  is 

b 

2a 
which  agrees  with  §  133. 

We  know  (§  133)  that  the  parabola  y  =  ax^  -\-bx-^  c  opens 
upward  or  downward  according  as  a  is  >  0  or  <  0.  Hence  the 
ordinate  of  the  vertex  is  a  minimum  ordinate,  i.e.  algebraically 
less  than  the  immediately  preceding  and  following  ordinates,  if 
a  >  0  ;  it  is  a  maximum  ordinate,  i.e.  algebraically  greater  than 
the  immediately  preceding  and  following  ordinates,  if  a  <  0. 

We  have  thus  a  simple  method  for  determining  the  max- 
imum or  minimum  of  a  quadratic  function  ax"^  -i-bx-^-  c;  the 
value  of  X  for  which  the  function  becomes  greatest  or  least  is 
found  by  equating  the  derivative  to  zero ;  the  quadratic  func- 
tion is  a  maximum  or  a  minimum  for  this  value  of  x  according 
as  a<  0  or  >  0.  ' 

Thus,  to  determine  the  greatest  rectangular  area  that  can  be  inclosed 
by  a  boundary  (e.g.  a  fence)  of  given  length  2  k,  let  one  side  of  the 


142  PLANE  ANALYTIC  GEOMETRY     [VIII,  §  140 

rectangle  be  called  x  ;  then  the  other  side  \^  k  —  x.     Hence  the  area  A  of 

the  rectangle  is 

A  =  x{k  —  .r)  =  kx  —  x^. 

Consequently  the  derivative  of  ^  is  k  —  2  x.  If  we  set  this  equal  to 
zero,  we  have  2x  =  k,  whence  x  —  k  12.  It  follows  that  k  —  x  —  k I2\ 
hence  the  rectangle  of  greatest  area  is  a  square 

EXERCISES 

1.  Locate  the  points  of  the  parabola  ?/  =  x-^  —  4  x  +  |  whose  abscissas 
are  —  1,  0,  1,  2,  3,  4,  draw  the  tangents  at  these  points,  and  then  sketch 
in  the  curve. 

2.  Sketch  the  parabolas  4  y  =  —  x'-^  +  4  x  and  ?/  =  x^  —  3  by  locating 
the  vertex  and  the  intersections  with  Ox  and  drawing  the  tangents  at 
these  points. 

3.  Is  the  function  y  =  5(x'-^  —  4  x  +  3)  increasing  or  decreasing  as  x 
increases  from  x  —  \'>    from  x  =  |  ? 

4.  Find  the  least  or  greatest  value  of  the  quadratic  functions  : 
(a)  2x-2-3x  +  6.         (6)8-6x-x2.  (c)x2-5x-5. 
(d)  2-2x-x2.            (e)4+x-^x'2.              (/)  5  x2  -  20x  +  1. 

5.  Find  the  derivative  of  the  linear  function  y  =  mx  -\-  h. 

6.  The  curve  of  a  railroad  track  is  represented  by  the  equation 
?/  =  I  x2,  the  axes  Ox,  Oy  pointing  east  and  north,  respectively  ;  in  what 
direction  is  the  train  going  at  the  points  whose  abscissas  are  0,  1,  2,  —  ^  ? 

7.  A  projectile  describes  the  parabola  y  =  jx—Sx^,  the  unit  being  the 
mile.  What  is  the  angle  of  elevation  of  the  gun  ?  What  is  the  greatest . 
height  ?     Where  does  the  projectile  strike  the  ground  ? 

8.  A  rectangular  area  is  to  be  inclosed  on  three  sides,  the  fourth  side 
being  bounded  by  a  straight  river.  If  the  length  of  the  fence  is  a  con- 
stant kj  what  is  the  maximum  area  of  the  rectangle  ? 

9.  Let  e  denote  the  error  made  in  measuring  the  side  of  a  square  of 
100  sq.  ft.  area,  and  E  the  corresponding  error  in  the  computed  area. 
Draw  the  curve  representing  E  as  ^  function  of  e. 

10.  A  rectangle  surmounted  by  a  semicircle  has  a  total  perimeter  of  100 
inches.  Draw  the  curve  representing  the  total  area  as  a  function  of  the 
radius  of  the  semicircle.     For  what  radius  is  the  area  greatest  ? 


VIII,  §  143] 


POLYNOMIALS 


143 


Fig.  56 


4 
18 


PART  II.     CUBIC   FUNCTION 

141.  The  Cubic  Function.  A  function 
of  the  form  aoX^  +  a^x^  +  ago;  +  ctg  is  called 
a  cubic  function  of  x.  The  curve  repre- 
sented by  the  equation 

y  =  aox^  -f  aiO^  -I-  a^x  +  dg 
can  be  sketched  by  plotting  it  by  points 
(§  131). 

For  example,  to  draw  the  curve  repre- 
sented by  the  equation 

y  z=:  OC^  —  2  X^  —  5  X  +  6, 

we  select  a  number  of  values  of  x  and  com- 
pute the  corresponding  values  of  y : 

a;=-3-2-101  2 

2/=- 24  0  860-4 

These  points  can  then  be  plotted  and  connected  by  a  smooth 
curve  which  will  approximately  represent  the  curve  corre- 
sponding to  the  given  equation  (Fig.  56). 

142.  Derivative.  The  sketching  of  such  a  cubic  curve  is 
again  greatly  facilitated  by  finding  the  derivative  of  the  cubic 
function;  the  determination  of  a  few  points,  with  their  tan- 
gents, will  suffice  to  give  a  good  general  idea  of  the  curve. 

To  find  the  derivative  of   the  function  y  =  aoX^  -f  aiO^  +  a.^ 
-f-ttg  the  process  of  §  138  should  be  followed.     The  student 
may  carry  this  out  himself;  he  will  find  the  quadratic  function 
y'  =  3  aifii^  -f-  2  ciiX  +  ag- 

143.  Maximum  or  Minimum  Values.  The  abscissas  of 
those  points  of  the  curve  at  which  the  tangent  is  parallel  to 
the  axis  Ox  are  again  found  by  equating  the  derivative  to 
zero;  they  are  therefore  the  roots  of  the  quadratic  equation 


144  PLANE  ANALYTIC  GEOMETRY    [VIII,  §  143 

3  a^  +  2  ajflj  +  cfca  =  0. 
If  at  such  a  point  the  derivative  passes  from  positive  to  nega- 
tive values,  the  curve  is  concave  doiv7iivard,  and  the  ordinate 
is  a  maximum;  if  the  derivative  passes  from  negative  to  posi- 
tive values,  the  curve  is  concave  upward,  and  the  ordinate  is 
a  minimum. 

144.  Second  Derivative.  The  derivative  of  a  function  of 
X  is  in  general  again  a  function  of  x.  Thus  for  the  cubic 
function  y  =  a^T?  +  aiX^  +  a^  +  a^  the  derivative  is  the  quad- 
ratic function  ^f  ^  3  ^^  ^2a,x  +  a,. 

The  derivative  of  the  first  derivative  is  called  the  second  deriva- 
tive of  the  original  function ;  denoting  it  by  y",  we  find  (§  139) 

?/"  =  6  (Xo^  +  2  ay. 
As  a  positive  derivative  indicates  an  increasing  function, 
while  a  negative  derivative  indicates  a  decreasing  function 
(§  140),  it  follows  that  if  at  any  point  of  the  curve  the  second 
derivative  is  positive,  the  first  derivative,  i.e.  the  slope  of  the 
curve,  increases ;  geometrically  this  evidently  means  that  the 
curve  there  is  concave  upward.  Similarly,  if  the  second  de- 
rivative is  negative,  the  curve  is  concave  downward.  We  have 
thus  a  simple  means  of  telling  whether  at  any  particular  point 
the  curve  is  concave  upward  or  downward. 

It  follows  that  at  any  point  where  the  first  derivative  van- 
ishes, the  ordinate  is  a  minimum  if  the  second  derivative  is 
positive ;  it  is  a  maximum  if  the  second  derivative  is  negative. 

145.  Points  of  Inflexion.  A  point  at  which  the  curve 
changes  from  being  concave  downward  to  being  concave  up- 
ward, or  vice  versa,  is  called  a  point  of  inflexion.  At  such  a 
point  the  second  derivative  vanishes. 

Our  cubic  curve  obviously  has  but  one  point  of  inflection, 
viz.  the  point  whose  abscissa  is  ic  =  —  ai/(3  a^). 


VIII,  §  145]  POLYNOMIALS  145 

EXERCISES 

1.  Find  the  first  and  second  derivatives  of  y  w^hen : 

(a)       ?/  =  6  x3  -  7  x2  -  X  +  2.         (6)     y  =  20  +  4  x  -  5  a;2  -  x^. 
(c)  10?/  =  x3-5x2+3x  +  9.  {d)      ?/  =  (x-l)(x-2)(x-3). 

(e)       2/  =  x2(x  +  3).  (/)  7?/  =  3x-2x(x2-l). 

2.  Sketch  the  curve  y  =  (x  —  2)  (x  +  1)  (x  +  3),  observing  the  sign  of  y 
between  the  intersections  with  Ox,  and  determining  the  minimum,  maxi- 
mum, and  point  of  inflection. 

3.  In  the  curve  y  =  acfifi  -\-  aix^  +  a2X  +  as,  what  is  the  meaning  of  as  ? 

4.  Sketch  the  curves  :  » 

(a)  5yz=(x-l)(x  +  4)2.  ,     (6)     y=(x-3)3. 

(c)  6  y  =  6  +  X  +  x2  -  x8.  (d)     y  =  x^-i  x. 

(e)  Sy  =  6 x2  -  x^.  (/)     y  =  x^  -  3 x2  +  4 x  -  5. 

5.  Draw  the  curves  y  =  x,  y  =  x^,  y  =  x^,  with  their  tangents  at  the 
points  whose  abscissas  are  1  and  —  1. 

6.  Find  the  equation  of  the  tangent  to  the  curve  14  y  =  5  x^  —  2  x2 
+x  —  20  at  the  point  whose  abscissa  is  2. 

7.  At  what  points  of  the  curve  y  =x^  —  ^x^  +  S  are  the  tangents 
parallel  to  the  line  ?/=— 3x+5? 

8.  Are  the  following  curves  concave  upward  or  downward  at  the 
indicated  points  ?     Sketch  each  of  them. 

(a)  y  =  4x3-6x,  atx  =  3.  (b)     3y  =  5x  -  3  x^,  at  x  =- 2. 

(c)  y  =  x3  -  2  x2  +  5,  at  X  =  i.       (c?)     2 y  =  x^  -  3 x2,  at  x  =  1. 

(e)  y  =  1 -x-x^,  atx  =  0.  (f)  10yz=x^+x^-l6x-{-6,a,tx=-^. 

9.  Show  that  the  parabola  y  =  ax^  +bx  -^  c  is  concave  upward  or 
concave  downward  for  all  values  of  x  according  as  a  is  positive  or  negative. 

10.  The  angle  between  two  curves  at  a  point  of  intersection  is  the 
angle  between  their  tangents.  Find  the  angles  between  the  curves  y  =  x^ 
and  y  =  x^  at  their  points  of  intersection. 

11.  Find  the  angle  at  which  the  parabola  y  =  2x2  —  3x  —  5  intersects 
the  curve  y  =  x^  4-  3  x  —  17  at  the  point  (2,  —  3). 

12.  The  ordinate  of  every  point  of  the  curve  y  =  x^  +  2  x2  is  the  sum  of 
the  ordinates  of  the  curves  y  =  x^  and  y  =  2x^.  From  the  latter  two 
curves  construct  the  former. 

L 


146 


PLANE  ANALYTIC  GEOMETRY    [VIH,  §  145 


13.  From  the  curve  y  =  x^  construct  the  following  curves : 

(a)  y=4:X^.        {b)  y  =  l- Y.       (c)  y  =  x^-2.        (d)  y  =  2x^  +  4. 


iij 


14.  Draw  the  curve  2y  =  x^  —  Sx^  and  its  reflection  in  the  line  y  =  x. 
What  is  the  equation  of  this  reflected  curve  ?  What  is  the  equation  of 
the  reflection  in  the  axis  Oy  ? 

15.  A  piece  of  cardboard  18  inches  square  is  used  to  make  a  box  .by 
cutting  equal  squares  from  the  four  corners  and  turning  up  the  sides. 
Draw  the  curve  whose  ordinates  represent  the  volume  of  the  box  as  a 
function  of  the  side  of  the  square  cut  out.     Find  its  maximum. 

16.  The  strength  of  a  rectangular  beam  cut  from  a  log  one  foot  in 
diameter  is  proportional  to  (i.e.  a  constant  times)  the  width  and  the 
square  of  the  depth.  Find  the  dimensions  of  the  strongest  beam  which 
can  be  cut  from  the  log.  Draw  the  curve  whose  ordinates  represent  the 
strength  of  the  beam  as  a  function  of  the  width. 

17.  Show  that  the  equation  of  a  curve  in  the  form  y  =  ax^  +  bx^  +  ex  +  d 
is  in  general  determined  by  four  points  Pi  (xi ,  yi),  Po  (X2 , 2/2),  Ps  (xs ,  ys), 
P*  (.Xi ,  ^4),  and  may  be  written  in  the  form 

y     x^     x^     X      1 
yi    Xi^    xi^    xi    1 

y2      X2^      X2^      X2      1 


ys    Xs'^    xs^    xs 


=  0. 


y^    Xa^    X4^    Xa 

18.  Find  the  equation  of  the  curve  in  the  form  y  =  ax^  -\-  bxJ^  -\-  cx  +  d 
which  passes  through  the  following  points  : 

(a)  (0,0),   (2,-1),  (-1,4),   (3,4); 
(6)  (1,1),  (3,-1),  (0,5),  (-4,1). 

19.  Show  that  every  cubic  curve  of  the  form  y  =  acfic^  +  aix^  +  a^x  +  a% 
is  symmetric  with  respect  to  its  point  of  inflection. 

146.   Cubic  Equation.     The  real  roots  of  the  cubic  equation 

a^  +  a^o^  +  ajOJ  +  ttg  =  0 
are  the  abscissas  of  the  points  at  which  the  cubic  curve 

?y  =  a^"^  4-  a^Q^  +  0^2^  +  ^3 
intersects   the   axis    Ox.     This   geometric   interpretation  can 


VIII,  §  146]  POLYNOMIALS  147 

be  used  to  find  the  real  roots  of  a  numerical  cubic  equation  ap- 
proximately :  calculating  *  the  ordinates  for  a  series  of  values 
of  X  (as  in  plotting  the  curve  by  points,  §  141),  or  at  least  deter- 
mining the  signs  of  these  ordinates,  observe  where  the  ordinate 
changes  sign.  At  least  one  real  root  must  lie  between  any 
two  values  of  x  for  which  the  ordinates  have  opposite  signs. 
The  first  approximation  so  obtained  can  then  be  improved  by 
calculating  ordinates  for  intermediate  values  of  x. 
Thus  to  find  the  roots  of  the  cubic 

a^4-ar^-16i»-f6  =  0 
we  find  that 

fora;  =  -5     -4-3-2-10        1        2        3       4 

2/ is  -        +         +        +++---     + 

The  roots  lie  therefore  between  —  5  and  —  4,  0  and  1,  3  and 
4.     To  find,  e.g.y  the  root  that  lies  between  0  and  1,  we  find  that 

for  a;  =  0     0.1     0.2     0.3     0.4 

2/is       +       +       +       +      - 
The  root  lies  therefore  between  0.3  and  0.4,  and  as  the  cor- 
responding values   of  y  are  1.317   and    —  0.176,  the   root  is 
somewhat  less  than  0.4.     As 

fora;=     0.40  0.39  0.38 

2/  =  -  0.176     -  0.029     +  0.119 
a  more  accurate  value  of  the  root  is  0.39. 

This  process  can  be  carried  as  far  as  we  please.  But  it  is 
very  laborious.  We  shall  see  in  a  later  section  (§  170)  how 
it  can  be  systematized. 

EXERCISES 

1.  Find  to  three  significant  figures  the  real  roots  of  the  equations : 
(a)  a;3  -  4 x2  +  6  =  0.  {h)   x^  +  x'^  -  x- \  =  0. 

(c)  a;3-3a:+l^=0.  (d)  x(x -l){x-2)=A. 


*  For  abridged  numerical  multiplication  and  division  see  the  note  on  p.  256. 


148  PLANE  ANALYTIC  GEOMETRY    [VIII,  §  147 

PART   III.     THE   GENERAL   POLYNOMIAL 

147.  Polynomials.  The  methods  used  in  studying  the 
quadratic  and  cubic  functions  and  the  curves  represented  by 
them  can  readily  be  extended  to  the  general  case  of  the  poly- 
nomial, or  rational  integral  function,  of  the  nth  degree, 

y  =  a^x"  +  a^sf"-^  +  a^vf"-^  H \-  a^_^x  +  a„ , 

where  the  coefficients  «„,  a^,  •••  a„  may  be  any  real  numbers, 
while  the  exponent  n,  which  is  called  the  degree  of  the  poly- 
nomial, is  a  positive  integer. 

We  shall  often  denote  such  a  polynomial  by  the  letter  y  or 
by  the  symbol  f{x)  (read  :  function  of  x,  or  /  of  a;) ;  its  value 
for  any  particular  value  of  x,  say  x  =  Xy  or  x  —  h,  is  then  de- 
noted by  /(iCi)  or  fQi),  respectively.  Thus,  for  x  =  0  we  have 
/(0)  =  a,. 

148.  Calculation  of  Values  of  a  Poljmomial.  In  plotting 
the  curve  y=f(x)  by  points  (§§  131,  141)  we  have  to  calculate 
a  number  of  ordinates.  Unless  f{x)  is  a  very  simple  poly- 
nomial this  is  a  rather  laborious  process.  To  shorten  it  ob- 
serve that  the  value  /(x^)  of  the  polynomial 

f(x)  =  aox""  -}-  ai«"-i  -f  • .  •  -\-a^ 
for  x  =  Xi  can  be  written  in  the  form 

f{xi)  =(  ...  {((aoXi  +  ai)xi-\-a2)Xi-i-as)Xi-\- \- a^_{)x-\- a,. 

To  calculate  this  expression  begin  by  finding  aQX^  -f  a^ ;  mul- 
tiply by  Xy  and  add  a^ ',  multiply  the  result  by  x^  and  add  a^  ; 
etc.     This  is  best  carried  out  in  the  following  form : 
Oo  %  ttg  •  •  •  ct,t 

Opa^  (a^Xy  -f  g^)  x^ 

a^flOi  -f  ai  (a^Xi  -f-  ai)Xi  +  ^g  •  •  • 
Eor  instance,  if 

f(x)  =  2  a^  -  3  «2  _  12  a;  +  5 

=  ((2a:-3)a;-12)a;-f-5, 


Vm,  §  149]  POLYNOMIALS  149 

to  find  /(3)  write  the  coefficients  in  a  row  and  place  2x3  =  6 
below  the  second  coefficient ;  the  sum  is  3.     Place  3  x  3  =  9  be- 
low the  third  coefficient ;  the  sum  is  —  3.    Place  3x(—  3)=  —  9 
below  the  last  coefficient;  the  sum,  —4,  is  =/(3). 
2-3-12         5 

6  9     -9 

2         3     _    3     _4 

This  process  is  useful  in  calculating  the  values  of  y  that  cor- 
respond to  various  values  of  x,  as  we  have  to  do  in  plotting  a 
curve  by  points.  It  is  also  very  convenient  in  solving  an  equa- 
tion by  the  method  of  §  146. 

EXERCISES 

1.  If  /(«)=  5x3  _  ISx  +  2,  what  is  meant  by  /(a)?  by  f{x  +  A)  ? 
What  is  the  value  of /(O)?  of /(2)  ?  of /(- 3  5)?  of/(-l)? 

2.  Find  the  ordinates  of  the  curve  y  =  x*  —  x^  +  3  x^  —  12  x  +  3  for 
X  =  3,  -  9,  -  i. 

3.  Find  the  ordinates  of  2  y  =  x*  +  3  x^  -  20  x  -  25  for  x  =  1,  2,  3,  -  1, 
-2.  ■ 

4.  Suppose  the  curve  y  =/(x)  drawn  ;  how  would  you  sketch  : 

(a)  2/=/(x-2)?     {b)   y  =  f(x+S)?    (c)  y  =  f(2x)?    (d)  y=f(-x)? 

(e)   y=f(^^y       if)y=f(x)+5?     (g)  y  =f(x-)-2x? 

6.   Calculate  to  three  places  of  decimals  the  real  roots  of -the  equations  : 
(a)  x3 -f  x2  =  100  ;   (&)  x^  -  4  =  0  ;  (c)  x^  -  7x  +  7  =  0. 

149.  Derivative  of  the  Polynomial.  We  have  seen  in  the 
preceding  sections  how  greatly  the  sketching  of  a  curve  and 
the  investigation  of  a  function  is  facilitated  by  the  use  of  the 
derivatives  of  the  function.  Thus,  in  particular,  the  first 
derivative  y'  is  the  rate  of  change  of  the  function  y  with  x, 
and  hence  determines  the  slope,  or  steepness,  of  the  curve 
y  =f{x).  We  begin  therefore  the  study  of  the  polynomial  by 
determining   its   derivative.     The   method   is   essentially  the 


150 


PLANE  ANALYTIC   GEOMETRY     [VIII,  §  149 


same  as  that  used  in  §§  138,  139  for  finding  the  derivative  of 
a  quadratic  function. 

The  first  derivative  y'  of  any  function  2/  of  a;  is  defined,  as 
in  §  138,  to  be  the  limit  of  the  quotient  Ay/ Ax  as  Aa;  approaches 
zero,  Ay  being  the  increment  of 
the  function  y  corresponding  to 
the  increment  Ax  of  a? ;  in  symbols : 


y' 


lim^, 
Ax=o  Aa; 


y 

y^  A  1              1 

^ 

//  1              I 
M   1               1 

X 

^     0 

/    N             .Q, 

A 

Fig.  57 

Geometrically  this  means  that  y' 

is  the  slope  of  the  tangent  of  the 

curve  whose  ordinate  is  y.     For,  Ay/ Ax  is  the  slope  of  the  secant 

PP,  (Fig.  57)  : 

— ^  =  tan  «! ; 

Ax  ' 

and  the  limit  of  this  quotient  as  Aa;  approaches  zero,  i.e.  as  P^ 
moves  along  the  curve  to  P,  is  the  slope  of  the  tangent  at  P: 


y'  =  tan  a  =  lim  ~ 


Aw 

im  -^ 

Ax=o  Aa; 


If  the  function  y  be  denoted  by  /(«),  then 
Ay=f(x-\-Ax)-f(x)', 


hence 


y 


^.^./XaH^A^WM^ 
Aa^  Aa; 


150.   Calculation  of  the  Derivative.     To  find,  by  means  of 
the  last  formula,  the  derivative  of  the  polynomial 

y  =f(^)  =  cto*"  +  «!«?"-'  +  •  •  •  +  a„, 
we  should  have  to  form  first /(a;  4-  Aa;),  i.e. 

(x  +  Axy  +  a,{x-hAxy-^-\-  ...  -fa„, 
subtract  from  this  the  original  polynomial,  then  divide  by  Aa;, 
and  finally  put  Aa;  =  0. 


VIII,  §  151]  POLYNOMIALS  •  151 

This  rather  cumbersome  process  can  be  avoided  if  we 
observe  that  a  polynomial  is  a  sum  of  terms  of  the  form  ax^ 
and  apply  the  following  fundamental  propositions  about 
derivatives : 

(1)  the  derivative  of  a  sum  of  terms  is  the  sum  of  the  deriva- 
tives of  the  terms  ; 

(2)  the  derivative  of  ax""  is  a  times  the  derivative  of  x"" ; 

(3)  the  derivative  of  a  constant  is  zero; 

(4)  the  derivative  of  x""  is  nx'^'K 

The  first  three  of  these  propositions  can  be  regarded  as 
obvious ;  a  fuller  discussion  of  them,  based  on  an  exact  defi- 
nition of  the  limit  of  a  function,  is  given  in  the  differential 
calculus.  A  proof  of  the  fourth  proposition  is  given  in  the 
next  article. 

On  the  basis  of  these  propositions  we  find  at  once  that  the 
derivative  of  the  polynomial 

y  =  ao.T"  -f-  aiO?""^  -f  OoX'^'^  4-  . . .  -f-  a,,^iX  +  a„ 
is 

y'  =  ao«a?"-^  +  «!  {n  —  1)  a;"-^  +  ag  {n  —  2)a7"-^  4-  •  •  •  +  a„_i  • 

151.  Derivative  of  ic»*.  By  the  definition  of  the  derivative 
(§  149)  we  have  for  the  derivative  of  y  =  x'': 

Aa^=0  Ax 

Now  by  the  binomial  theorem  (see  below,  §  152)  we  have 

(x  -f  Axy  =  .T'»  -h  nx^'-'^Ax  +  ^^i'^  —  ^)  a;»-2(Ax)-  +  ...  -f  (Aa;)% 
and  hence 

{x  -f  Axy  -  X"  =  nx''-^Ax-\-  ^(^'  ~  -*^)  x"-2(Aic)2  -|-  ...  -f  (Ax)\ 

Dividing  by  Ax  and  then  letting  Ax  become  zero,  we  find 
y'  =z  7ia;"~\ 


152  PLANE  ANALYTIC  GEOMETRY    [VIII,  §  151 

EXERCISES 

1.  Find  the  derivatives  of  the  following  functions  of  x  by  means  of 
the  fundamental  definition  (§  149)  and  check  by  §  150  : 

(a)  x\  (b)  x^  +  x.  (c)   x^+6x-^. 

(d)  -6x3.  (e)  a;* -3x3.  (y)  rnx -\- b. 

2.  Find  the  derivatives  of  the  following  functions  : 

(a)  5x4-3x2  +  6x.       (6)  l-x  +  ^x^-^x^        (c)   (x-2)3. 

(d)  (2x  +  3)5.  (e)  3(4^ X- 1)3.  (/)  x'»+ax"-i  +  6x"-2. 

3.  For  the  following  functions  write  the  derivative  indicated : 
(a)  5  x3  -  3  X,  find  y"'.  (&)  ax^  +  6x  +  c,  find  y'". 

(c)  x6,  find  y\  (^)  «^^  +  ^^^  +  ex  +  (Z,  find  y''\ 

(e)  ix6,  find  y".  (/)  ^a;6,  find  ?/-». 

(g)  xi2  _  g^xs,  find  y'".  (h)   (2  x  -  3)^,  find  y'". 

152.  Binomial  Theorem,     in  §  151  we  have  used  the  binomial 
theorem  for  a  positive  integral  exponent  w,  i.e.  the  proposition  that 

(1)       (x  +  A)"  =  x«  +  wa;«-iA  +  Vlllsi^x»-%^  +  ^(»-l)(y^-2)  ^.n-g^^a 

+  ...  +  ^iiLrA)-Ah-, 

n  ! 
The  formula  (1)  can  be  proved  by  mathematical  induction  (§  62).     It 
certainly  holds  for  n  =  2,  since  by  direct  multiplication  we  have 

(X  +  /i)2  =  x2  +  2x^1  +  A2  =  x2  +  2x^  +  ^h^, 

21 
which  agrees  with  (1)  for  n  =  2. 

Moreover,  if  the  formula  (1)  holds  for  any  exponent  w,  it  holds  for 
n  +  1.     For,  multiplying  (1)  by  x  -\-  h  in  both  members,  we  find 
(X  +  h)^+^  = 
xn+i  +  (n  +  l)xnh  +  (^  +  ^^^  X-1A2  +  ...  +  (n  +  l)n{n -1)  .••  1  ^^^^^ 

which  is  the  form  that  (1)  assumes  when  n  is  replaced  by  n  +  1. 

153.  Binomial  Coefficients.    The  coefficients 

2 !       '  (n-l)l  '  w ! 

in  the  binomial  formula  (1)  are  called  the  binomial  coefficients. 


VIII,  §  153]  POLYNOMIALS      '  153 

The  meaning  of  these  coefficients  will  appear  from  another  proof  of  the 
formula,  which  is  as  follows :  If  n  is  a  positive  integer,  we  can  write 
(x  +  y^)"  in  the  form 

(x  +  hi){x  +  h2){x  +  hz){x  +  hi)  ...  {X  +  /i„), 

where  the  subscripts  are  used  simply  for  convenience  to  distinguish  the 
binomial  factors ;  i.e.  it  is  understood  that  hi  =  h2  =  hz=  •••  =  hn=  h. 
Each  term  in  the  expanded  product  is  the  product  of  n  letters  of  which 
one  and  only  one  is  taken  from  each  binomial  factor.  To  form  all  these 
terms  we  may  proceed  as  follows  : 

(a)  If  we  choose  x  from  each  of  the  n  factors,  we  obtain  as  first  term 
of  the  expansion  x^. 

(b)  If  we  choose  x  from  n  —  1  factors,  the  letter  h  can  be  chosen  from 
any  one  of  the  n  factors,  i.e.  it  can  be  chosen  in  „(7i  ways  (§  64)  ;  this 
gives 

x'*-'^(hi  4-  ^2  +  •••  +  ^n)i  the  number  of  terms  being  „Ci. 

(c)  If  we  choose  x  from  n  —  2  factors,  the  other  two  letters  can  be 
chosen  from  any  two  of  the  n  factors,  i.e.  in  ^(h  ways  ;  this  gives 

x^~^(hih2  +  hihs  4-  •••  +  ^2^3  +  •••)»  ^^^  number  of  terms  being  „CV 

(d)  If  we  choose  x  from  n  —  3  factors,  the  other  three  letters  can  be 
chosen  from  any  three  of  the  n  factors,  i.e.  in  nCs  ways  ;  this  gives 

x>'-^(hihihz-\-hxh2h^  +  •••  +^2^3^4  +  •-•)»  ^^^  number  of  terms  being  ^Cg. 

Finally  we  have  to  choose  no  x  and  consequently  an  h  from  every  factor, 
which  can  be  done  in  „C„=1  way  ;  this  gives  the  last  term 

hihi  -'  K. 

Now  as  ^1  =  A2  =  •••  =h„=h,  we  find  the  binomial  expansion  : 

(X  +  h)^  =  a:«  +  nCiX'*-!^  +  rtCiX^-^h^  +   •••   +  nCn-lXh^'^  +  nCrM. 

Since,  by  §  64, 

1   •  Ji 

this  form  agrees  with  that  of  §  152.  It  will  now  be  clear  why  the 
binomial  coefficients  are  the  numbers  of  combinations  of  n  elements, 
1,  2,  3,  •••  at  a  time. 


154  PLANE  ANALYTIC  GEOMETRY    [VIII,  §  153 

The  proof  also  shows  that  the  binomial  coefficients  are  equal  in  pairs, 
the  first  being  equal  to  the  last,  the  second  to  the  last  but  one,  etc. 

Finally  it  may  be  noted  that,  with  cc  =  1,  ^  =  1  we  obtain  the  following 
remarkable  expression  for  the  sum  of  the  binomial  coefficients : 

EXERCISES 

1.  Show  that  in  the  expansion  of  (x— ^)'*  by  the  binomial  theorem  the 
signs  of  the  terms  are  alternately  +  and  — . 

2.  If  the  binomial  coefficients  of  the  first,  sec-  1 

ond,  third,  fourth,  etc.,  power  of  a  binomial  are  1      1 

12      1 
written  down  as  in  the  horizontal  lines  of  the 

adjoining  diagram,  it  will  be  observed  that  (ex-  14    6     4     1 

cepting  the  ones)  every  figure  is  the  sum  of  the  1    5    10    10    5    1 

two  just  above  it.    Extend  the  triangle  by  this  rule 

to  the  10th  power,  and  prove  the  rule  (see  §  152).  Pascal's  Triangle 

3.  Expand  by  tne  binomial  theorem  : 

(a)  {x  +  2y)^  '     (6)   (^'+1)'-  (c)    (2a-c)3. 

00   (--^X-  («)    (a  +  b  +  cy.       (/)  {4:x-lyy. 

\y    x^J 

(g)    (H-2x)3-(l-2x)3.      (^h)   (l^xy^  (0     (^-^)*'- 

(.0    (f-^-l)'-  W   (|--a^2)*.  (0     (a  +  b-c-dy. 

4.  Write  the  term  indicated  : 

(a)   Fourth  term  in  (a  4-  by^.  (d)    Middle  term  in  (x^  —  y^y^. 

(6)   Fourth  term  in  (a  -  by^.  (e)     A;th  term  in  (x  +  hy. 

(c)    Tenth  term  in  (x^  -f  4  y^y^.       (/)    kth  term  in  (x  -  hy. 
(g)   Two  middle  terms  in  (a^  _  2  b'^y. 

(1  \2'^ 
a  —  ]    . 

5.  Show  that  the  sum  of  the  coefficients  in  the  expansion  of  {x—hy  is 
zero  when  n  is  an  odd  integer. 

6.  Use  the  binomial  formula  to  find     {a)   (1.02)6;    (5)   (3.97)«. 


VIII,  §  155]  POLYNOMIALS  155 

154.  Properties  of  the  General  Polynomial  Curve.  In  plot- 
ting the  curve 

y  =  OoX"  4-  ajic""^  -h  a2^"~^  H-  ••  •  +  «„ 

observe  that  (Fig.  58)  : 

(a)  the  intercept  OB  on  the  axis  Oy 
is  equal  to  the  constant  term  a„ ; 

(h)  the  intercepts  OA^,  OA^,  •••on 
the  axis  Ox  are  roots  of  the  equation 
y  =  0,  i.e.  '    Fig.  58 

(c)  the  abscissas  of  the  least  and  greatest  ordinates  are 
found  by  solving  the  equation  y'  =  0,  i.e.  (§  150) 

every  real  root  giving  a  minimum  ordinate  if  for  this  root  y" 
is  positive  and  a  maximum  ordinate  if  y"  is  negative ; 

(d)  the  abscissas  of  the  points  of  inflection  are  found  by 
solving  the  equation  y"  =^0,  i.e. 

n(w-l)ao^"-2+  ...  +2a„_2  =  0, 

every  real  root  of  this  equation  being  the  abscissa  of  a  point 
of  inflection  provided  that  y"'=^0.  (If  y'"  were  zero,  y'  might 
not  be  a  maximum  or  minimum,  and  further  investigation 
would  be  necessary.) 

155.  Continuity  of  Polynomials.  It  should  also  be  ob- 
served that  the  function  y  =  a^pf  +  Oja;""^  +  —  +  a„  is  one- 
valued,  real,  and  finite  for  every  x ;  i.e.  to  every  real  and  finite 
abscissa  x  belongs  one  and  only  one  ordinate,  and  this  ordinate  is 
real  and  finite.  Moreover,  as  the  first  derivative  y'  =  noojc""^ 
+  •••  +«„_!  is  again  a  polynomial,  the  slope  of  the  curve  is 
everywhere  one-valued  and  finite. 


156 


PLANE  ANALYTIC  GEOMETRY     [VIII,  §  155 


Thus,  so-called  discontinuities  of  the  ordinate  (Fig.  59)  or  of 
the  slope  (Fig.  60)  cannot  occur :  the  curve  y  =  a^''  4-  —  -\-  a„ 
is  continuous. 


Strictly  defined,  the  continuity  of  the  function  y  =  a^""  +  — 
4-  a„  means  that,  for  every  value  of  x,  the  limit  of  the  function 
is  equal  to  the  value  of  the  function.  The  function  y  =  a^fid^  +  ••• 
4-a„  has  one  and  only  one  value  for  any  value  x  =  x^j  viz. 
(^1  -\-  •'•  +^n-  The  value  of  the  function  for  any  other 
value  of  X,  say  for  oci  +  Ace,  is  a^i(Xl  -f  Aa?)'*  +  —  +  a„  which  can 
be  written 'in  the  form  aQXi" -\-  —  +a„-l- terms  containing  Aa; 
as  factor.  Therefore  as  Aa;  approaches  zero,  the  function 
approaches  a  limit,  viz.  its  vahie  for  x=:Xi. 

156.  Intermediate  Values.  A  continuous  function,  in 
varying  from  any  value  to  any  other  value,  must  necessarily 
pass  through  all  intermediate  values.  Thus,  our  polynomial 
y  =  a(fic''  -f  ..•  -f  a„,  if  it  passes  from  a  negative  to  a  positive 
value  (or  vice  versa),  must  pass  through  zero.  It  follows 
from  this  that  heticeen  any  two  ordinates  of  opposite  sign  the 
curve  y  =  aoX'*  +  •••  +  ^„  must  cross  the  axis  Ox  at  least  once. 

It  also  follows  from  the  continuity  of  the  polynomial  and 
its  derivatives  that  between  any  two  intersections  ivith  the  axis 
Ox  there  must  lie  at  least  one  maximum  or  minimum^  and  be- 
tween a  maximum  and  a  minimum  there  must  lie  a  point  of 
inflection. 

Ordinates  at  particular  points  can  be  calculated  by  the  pro- 
cess of  §  148. 


VIII,  §  156]  POLYNOMIALS  157 

EXERCISES 

1.  Sketch  the  following  curves  : 

(a)  y=(x-l)ix-2){x-S).       (&)  4i/  =  x4-l.  (c)  10y  =  x^. 

(d)  10y  =  x^-\-5.  (e)  iy={x-h2)%x^S).     (/)  y={x-lY. 

2.  When  is  the  curve  y  =  aox**  +  aix»-i  +  •••+««  symmetric  with 
respect  to  Oy  ? 

3.  Determine  the  coefficients  so  that  the  curve  y  =  aox'^  +  aix^  +  azx^ 
+  a^x  +  a4  shall  touch  Ox  at  (1,  0)  and  at  (—  1,  0)  and  pass  through 
(0,  1),  and  sketch  the  curve. 

4.  Find  the  coordinates  of  the  maxima,  minima,  and  points  of  inflec- 
tion and  then  sketch  the  curve  4  ^  =  x"*  —  2  x^. 

5.  Are  the  following  curves  concave  upward  or  downward  at  the  indi- 
cated points  ? 

(a)  16^  =  16x4-8x2  +  1,  atx=-l,  -  |,  0,  |,  3. 
■  (6)        y=4:X-xS  at  X  =  -  2,  0,  1,  3. 

(c)  y  =  X",  at  any  point ;  distinguish  the  cases  when  n  is  a  positive 
even  or  odd  integer. 

6.  What  happens  to  the  curves  y  =  ax^  and  y  =  ax^  as  a  changes  ? 
For  example,  take  a  =  2,  1,  ^,  0,  —  |,  —  1,  —  2. 

7.  Find  the  values  of  x  for  which  the  following  relations  are  true  : 
(a)  x*  -  6  x2  +  9  ^  0.  (6)   (x  -  l)2(x2  -  4)  ^  0. 

8.  Show  that  the  following  curves  do  not  cross  the  axis  Ox  outside  of 
the  intervals  indicated : 

(a)  ?/  =  x^  —  2  x2  4-  4x  +  5,  between  —  2  and  2. 
(&)  i/  =  x4-5x2  +  6x-3,    -3  and  3. 

(c)  y  =  x3-x2  +  3x-3,  Oand  1. 

(d)  y  =  x4  +  x2  -  3  X  +  2,   0  and  1. 

9.  Those  curves  whose  ordinates  represent  the  values  of  the  first, 
second,  etc.,  derivatives  of  a  given  polynomial  are  called  the  first,  second, 
etc.,  derived  curves.  Sketch  on  the  same  coordinate  axes  the  following 
curves  and  their  derived  curves  : 

(a)  6^=2x3-3x2- 12x.  (b)      y  =  (x- 2y(x  +  1). 

(c)     y  =  (x+  1)3.  (d)  2  y  =  X*  +  x2  +  1. 

10.  At  what  point  on  Ox  must  the  origin  be  taken  in  order  that  the 
equation  of  the  curve  y  =  2x^  —  Sx^  —  12x  —  5  shall  have  no  term  in  x2  ? 
no  term  in  x  ? 


158  PLANE  ANALYTIC  GEOMETRY    [VHI,  §  157 

PART   IV.     NUMERICAL   EQUATIONS 

157.  Equations.  Roots.  In  plotting  the  curves  y  —  afpif  + 
•••  +  ^n  (§  ^^^)  i*  is  often  desirable  to  solve  equations  of  the  form 

(1)  ao^;"  +  -  +  «„  =  0, 

the  coefficients  «o>  %>  ••*  ««  being  given  real  numbers  and  n  any 
positive  integer.  The  solution  of  such  numerical  equations^ 
at  least  approximately,  presents  itself  in  many  other  prob- 
lems. The  roots  of  the  equation  (1)  are  also  called  the  roots, 
OY  zeros,  of  the  function  a^fc'' -\-  •••  +ot„- 

It  is  understood  that  a^  4^  0  since  otherwise  the  equation 
would  not  be  of  degree  n.  We  can  therefore  divide  (1)  by  a^ 
and  write  the  equation  in  the  form 

(2)  x^+x>^:ff-^-^  ...  H-i),=0, 

where  p^=ia^/aQ,  i>2=«2/«o?  —  i>,»  =  «„/^o  are  given  real  numbers. 

158.  Relation  of  Coefficients  to  Roots.  A\^e  here  assume 
the  fundamental  theorem  of  algebra  that  every  equation  of  the 
form  (2)  has  at  least  one  root,  say  x  =  x^,  which  may  be  real  or 

imaginary.     If  we  then  divide  the  polynomial  x^-\-p^x''~^-\ \-p^ 

by  a;  —  Xy,  we  bbtain  a  polynomial  of  degree  ii  —  1 ;  the  equation 
of  the  (n  —  l)th  degree  obtained  by  equating  this  polynomial  to 
zero  must  again  have  at  least  one  root.  Proceeding  in  this 
way,  we  find  that  every  equation  of  the  form  (2)  has  n  roots, 
which  of  course  may  be  real  or  complex,  and  some  of  which 
may  be  equal.  It  also  appears  that  the  equation  (2)  may  be 
written  in  the  form 

(3)  {x  -  x^){x  -x^"'{x-  a;„)  =  0, 

where  x^,  x^,  •.«  x^  are  the  n  roots,  or  performing  the  multiplica- 
tion (§  153) : 

(4)  a;"  -  (a?!  +  . . .  -f  x„)a;"-i  -f-  {x^x.  +  •  •  •  +  x^^^x^x"""^  -f  •  • . 

-|-(— 1)%  ...  05^  =  0. 


VIII,  §  159]  NUMERICAL  EQUATIONS  159 

Comparing  the  coefficients  in  (4)  with  those  in  (2),  we  find : 
Xi  +  -"+Xn  =  -pi, 


i.e.  if  the  coefficient  of  the  highest  power  of  a  polynomial  is 
one,  then  the  coefficient  of  a;"~^,  with  sign  reversed,  is  equal  to 
the  sum  of  the  roots;  the  coefficient  of  x""'^  is  equal  to  the  sum 
of  the  products  of  the  roots  two  at  a  time  ;  minus  the  coefficient 
of  «'*"'  is  equal  to  the  sum  of  the  products  of  the  roots  three  at 
a  time,  etc. ;  plus  or  minus  the  constant  term  (according  as  n  is 
even  or  odd)  is  equal  to  the  product  of  all  the  roots. 

159.  Equations  with  Integral  Coefficients.  The  results  of 
the  last  article  can  often  be  used  to  advantage  to  find  the  roots 
of  a  numerical  equation  (2)  in  which  all  the  coefficients  pi,  "-p^ 
are  integers.  We  then  try  to  resolve  the  left-hand  member 
into  linear  factors  of  the  form  x  —  x,^;  if  this  can  be  done,  the 
roots  are  the  numbers  x^. 

The  fact  that  the  constant  term  j)^  in  (2)  is  plus  or  minus 

the  product  of  the  roots  can  be  used  in  the  same  case  by  trying 

to  see  whether  any  one  of  the  integral  factors  of  ±  p^  satisfies 

the  equation. 

EXERCISES 

1.  Findtherootsof  :  (a)  x^  -  7  a;-}-6=0  ;  (&)  a;3-2  x2-13a;-10=0  ; 
(c)  x*  -  1  =  0  ;  (d)  x*  -  7  x2  -  18  =  0  ;  (e)  a;^  -  5  a;2  -  2  x  +  24  =  0. 

2.  Form  the  equation  whose  roots  are  :  (a)  2,  —  2,  3  ;  (&)  —  1,  —  1,  1  ; 
(c)  0,  V2,  -V2;  (d)  -1,  1,  i,  -f. 

3.  For  the  equation  x^  +  pix^  +  pox  -\-p3  =  0  determine  the  relation 
between  the  coefficients  when  :  (a)  two  roots  are  equal  but  opposite  in 
sign ;  (6)  the  product  of  two  roots  is  equal  to  the  square  of  the  third  ; 
(c)  the  three  roots  are  equal. 

4.  Show  that  the  sum  of  the  n  nth  roots  of  any  number  is  zero.  What 
about  the  sum  of  the  products  of  the  roots  two  at  a  time  '?  three  at  a  time  ? 


160  PLANE  ANALYTIC  GEOMETRY    [VIII,  §  160 

160.  Imaginary  Roots.  In  general,  the  real  roots  of  a 
numerical  equation  are  of  course  not  integers,  nor  even  rational 
fractions,  but  irrational  numbers.  In  solving  such  an  equation 
the  object  is  to  find  a  number  of  decimal  places  of  each  root 
sufficient  for  the  problem  in  hand.  Methods  of  approximation 
appropriate  for  this  purpose  are  given  in  the  following  articles. 

The  imaginary  roots  of  the  equation  can  be  determined  by 
somewhat  similar,  though  more  laborious,  processes.  It  will 
here  suffice  to  show  that  imaginary  roots  always  occur  in  pairs 
of  conjugates  ;  that  is,  if  an  imaginary  number  a  -\-  pi  is  a  root 
of  the  equation  (1)  (with  real  coefficients),  theii  the  conjugate 
imaginary  number  a  —  /Si  is  a  root  of  the  same  equation. 

For,  substituting  a  -j-  ^i  for  x  in  (1)  and  collecting  the  real 
and  pure  imaginary  terms  separately,  we  obtain  an  equation  of 
the  form  A-{-Bi  =  0, 

where  A  and  B  are  real ;  hence,  by  §  116,  ^  =  0  and  B  =  0. 

If,  on  the  other  hand,  we  substitute  in  (1)  a  —  /3^  for  x,  the 
result  must  be  the  same  except  that  i  is  replaced  by  —  i;  we 
find  therefore  A  —  Bi  =  0,  and  this  is  satisfied  if  A  =  0  and 
^  =  0,  i.e.  if  a  -h  (ii  is  a  root. 

It  follows  in  particular  that  a  cubic  equation  always  has  at 
least  one  real  root.  Indeed,  in  the  case  of  the  cubic  equation, 
only  two  cases  are  possible :  (a)  the  equation  has  three  real 
roots,  which  may  of  course  be  all  different,  or  two  equal  but 
different  from  the  third,  or  all  three  equal ;  (b)  the  equation 
has  one  real  and  two  conjugate  imaginary  roots. 

161.  Methods  of  Approximation  for  Real  Roots.  If  a  good 
sketch  of  the  curve  y  =  aQX''-\-,  •••  +  a„  were  given,  we  could 
obtain  approximate  values  of  the  real  roots  of  the  equation 

ao'K"  4-  •••  +a^  =  0 
by  measuring  the  intercepts  OA^j  OA^,  etc.,  made  by  the  curve 


VIII,  §  162] 


NUMERICAL  EQUATIONS 


161 


on  the  axis  Ox  (§  154).  If  the  curve  is  not  given,  we  calculate 
a  number  of  ordinates  for  various  values  of  x  until  we  find 
two  ordinates  of  opposite  sign  ;  we  know  (§  156)  that  the  curve 
must  cross  the  axis  Ox  between  these  ordinates,  and  therefore 
at  least  one  real  root  of  the  equation  must  lie  between  the 
abscissas,  say  x^  and  x^,  whose  ordinates  are  of  opposite  sign. 

We  can  next  contract  the  interval  between  which  the  root 
lies  by  calculating  intermediate  ordinates.  By  this  process  a 
root  can  be  calculated  to  any  desired  degree  of  accuracy.  But 
the  process  is  rather  long  and  laborious.  The  calculation  of 
the  ordinates  is  best  performed  by  the  process  of  §  148. 


162.  Interpolation.  If  the  interval  within  which  the  root 
has  been  confined  is  small,  we  can  obtain,  without  calculating 
further  ordinates,  a  further  approximation  to  the  root  by 
replacing  the  curve  in  the  interval  by  its  secant,  and  finding 
its  intersection  with  the  axis  Ox. 

Suppose  (Fig.  61)  that  we  have 
found  that  a  root  lies  between 
OQy  =  Xy  and  0Q2  =  X2,  the  ordi- 
nates QiPi  =  2/1  and  Q2P2  =  2/2  being 
of  opposite  sign.  Then  Xi  is  ^.  first 
approximation  to  the  root  a;;  and 
if  Qi  and  Q2  lie  close  together,  the 
intercept  OQ  made  by  the  secant 

PjPo  on  the  axis  Ox  is  a  second  approximation.  Let  us 
calculate  the  correction  Q^Q  =  h  which  must  be  added  to 
the  first  approximation  x^  to  obtain  the  second  approximation 
x^  +  h. 

The  figure  shows  that  QiQ/BP2  =  PiQi/PiR,  i-e. 


Fig.  61 


2/2-2/1 


162  PLANE  ANALYTIC   GEOMETRY     [VIII,  §  162 

hence  the  correction  li  is 


7i  = 


Out)  — ~  tJC-i  LAtJu 

^  2/1  =  -  —2/1- 

Vi  -  Vi  ^y 


This  process,  which  is  the  same  as  that  used  in  interpolating 
in  a  table  of  logarithms,  is  known  as  the  regula  falsi,  or  rule 
of  false  position. 

163.  Tangent  Method.  Another  method  for  finding  a  correction 
consists  in  using  the  intercept  made  on  the  axis  Ox  not  by  the  secant  but 
by  the  tangent  to  the  curve  at  Pi. 

The  correction  Qi  Q'  =  k  is  found 
(Fig.  61)  from  the  triangle  PiQiQ',  in 
which  the  tangent  of  the  angle  at  Q'  is 
equal  to  the  value  of  the  derivative  yi' 
at  Pi.     This  triangle  gives 


k 


hence  k=—  ^-  . 

y\' 


yi  . 
k' 


Fig.  61 


Find  by  this  method  the  roots  of  0^  —  305+1=0. 


164.   Newton's  Method  of  Approximation.     After  finding, 
by  §  161,  a  first  approximation  x^  to  a  root  of  the  equation 

(1)  ao^"  +  a,x--'  +  . . .  +  a„  =  0, 

transfer  the  origin  to  the  point  (xi,  0).      Thus  (Fig.  62),  if  a 
root  lies  between  3  and  4,  transform  the 
equation  to  (3,  0)  as  origin,  by  replacing 
a;  by  3  -h  h.     An  expeditious  process  for 
finding  the  new  equation  in  h,  say 


(2)        6o/i«4-M"-' +  •••  +  &„=  0, 
will  be  given  in  §§165-167. 


Fig.  62 


VIII,  §  166]  NUMERICAL  EQUATIONS  163 

As  li  is  a  proper  fraction,  its  higher  powers  will  be  small, 
so  that  an  approximate  value  of  h  can  be  obtained  from  the 
linear  terms,  i.e.  by  solving  h,,_^h-\-h,^  =0,  which  gives  h  ap- 
proximately =  —  6„  /  6„_i.     Hence  we  put 

(3)  h  =  -^-^k, 

where  A:  is  a  still  smaller  proper  fraction.  If  the  approxima- 
tion obtained  from  the  linear  terms  should  be  too  rough,  we 
may  find  a  better  approximation  of  h  by  solving  the  quadratic 

K-2h'+K_,h  +  K  =  0. 
We  next  substitute  the  value  (3)  of  h  in  (2)  and  proceed  in 
the  same  way  with  the  equation  in  k.  The  process  can  be 
repeated  as  often  as  desired ;  the  last  division  can  be  carried 
to  about  as  many  more  significant  figures  as  have  been  obtained 
before.     The  example  in  §  168  will  best  explain  the  work. 

165.  Remainder    Theorem,     if    a  polynomial  f(x)  =  aox»  + 

aix^-^  4-  • .  •  +  a„  of  degree  n  be  divided  by  x  —  h,  there  is  obtained  in 
general  a  quotient  Q,  which  is  a  polynomial  of  degree  n  —  I,  and  a  re- 
mainder B : 

^=  Q  +  ^,  i.e.   fix)  =  Q(x-h)  +  B. 
X  —  h  X  —  h 

For  X  =  h  the  last  equation  gives /(^)  =  B  ;  i.e.  the  value  of  the  poly- 
nomial for  any  particular  value  h  of  x  is  equal  to  the  remainder  B  ob- 
tained upon  dividing  the  polynomial  by  x  —  h : 

fih)  =  aoh^  +  -'--\-an  =  B. 

This  proposition  is  known  as  the  remainder  theorem. 

166.  Synthetic  Division.    As  an  example  let  us  divide 

f(x)  =  2  x3  -  3  ic2  -  12  a;  -I-  5 
"by  a;  —  3.     By  any  method  we  obtain  the  following  result  : 

X  —  3  x—  S 


164  PLANE  ANALYTIC  GEOMETRY    [VIII,  §  166 

The  elementary  method  is  as  follows  : 

2  a;8  -  6  x^ 

3  x2  -  12  X 
3a;2_   9a; 


-3x  +  5 
-  3  X  +  9 


-4 

This  process  can  be  notably  shortened  : 

(a)  As  the  dividend  is  a  polynomial,  it  can  be  indicated  sufficiently  by 
writing  down  its  coefficients  only,  any  missing  term  being  supplied  by  a 
zero ;  2     -  3     -  12    5 

(5)  As  X  in  the  divisor  has  the  coefficient  1,  the  first  terms  of  the 
partial  products  need  not  be  written ;  the  second  terms  it  is  more  con- 
venient to  change  in  sign  ;  in  other  words,  instead  of  multiplying  by  —  3 
and  subtracting,  multiply  by  +  3  and  add. 

The  whole  calculation  then  reduces  to  the  following  scheme : 
2-3-12         5[3 

6  9-9 

2         3-3-4 

This  is  the  same  scheme  as  that  in  §  148.  But  it  should  be  observed 
that  this  method,  known  as  synthetic  division,  gives  not  only  the  remain- 
der —  4,  i.e.  /(3),  but  also  the  coefficients  2,  3,  —  3  of  the  quotient. 

167.    Calculation  of  /(aj^  +  h).    if  in  /(x)  =  aox«  +  •  •  •  +  «„  we  sub- 
stitute x  =  Xi-{-h,  we  find : 
/(x)  =/(xi  +  h)  =  ao(xi  +  h)^  +  ai(xi  +  A)«-i  +  ••.  +  a«-i  (xi  +  h)  +  a„. 

Expanding  the  powers  of  Xi  4-  A  by  the  binomial  theorem  and  arrang- 
ing in  descending  powers  of  h  we  obtain  a  result  of  the  form 

/(x)  =/(xi  +  h)=b^^  +  hih^-^  +  •••  4-  bn.ih  +  6„. 

To  find  the  coefficients  6o,  6i,---  &„  of  this  expansion  of  /(xi  +  h)  in 
powers  of  h  observe  that  as  A  =  x  —  Xi  we  have 
/(x)  =/(xi  +  h)  =  bo{x  -  xi)«  4-  l>i(x -^  xi)"-!  +  ••.  +  6„_i(x  -  xi)  +  &«. 

The  last  term,  6„,  is  therefore  the  remainder  obtained  upon  dividing 
f(x)  by  X—  xi ;  it  is  best  found  by  synthetic  division  (§  166).  The  quo- 
tient obtained  upon  dividing  /(x)  by  x  —  Xi  is  evidently  6o(x  — xi)""^ 
+  &i(x  —  xi)''-^  4-  •••  4  6„_i ;   the  last  term,  6„_i,  can  again  be  obtained 


VIII,  §  168]  NUMERICAL  EQUATIONS  165 

as  the  remainder  upon  division  by  x  —  xi.    Proceeding  in  this  way  all 
the  coefiBcients  6„,  b^-u  •••  &i,  &o  can  be  found. 
For  the  example  of  §  166  we  have 

2-3-12         5|3 


6 

9 

-9 

2 

3 

-3 

-4 

6 

27 

2 

9 
6 

24 

2         15 

The  result  is :     f{S-{-h)=2h^  +  15h'^  +  2^h-  4. 

168.    Example.     The  roots  of  the  equation 

2  a;3  _  3  a;2  _  12  X  +  5  =  0 

are  readily  found  to  lie  between  —  3  and  —  2,  0  and  1,  3  and  4.  To 
calculate  the  last  of  these  we  find  by  transferring  the  origin  to  the  point 
(3,  0)  the  following  equation  for  the  correction  h  to  the  first  approxima- 
tion, which  is  3  (§  167)  : 

The  linear  terms  give  h  =  1/6  =  0.17;  as  the  quadratic  term,  15  h^,  is 
about  0.42  and  1/24  of  this  is  0.02,  a  somewhat  better  approximation  is 
h  =  0.15.     Substituting 

Ji  =  0.16  + hi, 
we  find:  2     15  24  -4 

0.30    2.295     3.94425 
2  15.30   26.295   -0.05575 

.30    2.340 
2  15.60   28.635 

^ 

2     15.90 
Hence  the  equation  for  hi  is 

2  hi^  +  15.90  hi^  +  28.635  hi  -  0.06575  =  0. 
The  linear  terms  give  ^   =  0  001947 

As  the  quadratic  term  can  influence  only  the  6th  decimal  place,  we  can 
certainly  take  ^1  =  0.00195  and  thus  find  the  root  3.15195. 


166  PLANE  ANALYTIC   GEOMETRY    [VIII,  §  169 

169.  Negative  Roots.  To  find  a  negative  root  replace  a;  by  —  x 
in  the  given  equation,  i.e.  reflect  the  curve  in  the  axis  Oy. 

To  find  a  root  greater  than  10  replace  x  by  10 «,  or  100  2;,  etc.,  in  the 
given  equation,  and  calculate  z. 

170.  Horner's  Process.  W.  G.  Homer's  method  is  essentially  the 
same  as  Newton's,  inasmuch  as  it  consists  in  moving  the  origin  closer 
and  closer  up  to  the  root.  But  it  calculates  each  significant  figure 
separately.     Thus,  for  the  example  of  §  168  we  should  proceed  as  follows: 

As  in  §§  167, 168,  we  diminish  the  roots  of  the  equation 

2x3-3x2_i2x  +  5  =  0 

by  3  so  that  the  equation  (as  there  shown)  takes  the  form 

2  a;3  +  15  a:2  +  24  X  -  4  =  0. 

The  left-hand  member  changes  sign  between  0.1  and  0.2.     We  move  there- 
fore the  origin  through  0.1  to  the  right : 

2     15  24          -4 

.2  1.52         2.552 

2     15.2  25.52     -1.448 

.2  1.54 

2    16.4  27.06 

.2 


2     15.6 


The  new  equation  is  2  x^  +  15.6  x^  +  27.06  x  -  1.448  =  0. 

The  left-hand  member  changes  sign  between  0.05  and  0.06  ;  hence  we 
move  the  origin  through  0.05 : 

2     15.6  27.06  -1.448 

.10        .785         1.39225 

2     15.70  27.845  -0.05576 

.10        .790 


16.80    28.636 
.10 


2     15.90 

The  new  equation  is  2  x^  +  15.90  x^  +  28.636  x  -  0.05575  =  0. 

We  can  evidently  go  on  in  the  same  way  finding  more  decimal  places. 
It  should  not  be  forgotten  (§  164)  that  after  finding  a  number  of  significant 


VIII,  §  170]  NUMERICAL  EQUATIONS  167 

figures  in  this  way,  about  as  many  more  can  be  found  by  simple  division. 
Thus,  we  have  found  x  =  3.15  •••  ;  the  linear  terms  of  the  last  equation 
give  the  correction  0.00195,  so  that  x  =  3.15195. 

EXERCISES 

1.  Find:  (a)  the  cube  root  of  67 ;  (&)  the  fourth  root  of  19 ;  (c)  the 
fifth  root  of  7,  to  seven  significant  figures,  and  check  by  logarithms, 

2.  Newton  used  his  method  to  approximate  the  positive  root  of 
x^  —  2ic  —  5=0;  find  this  root  to  eight  significant  figures. 

3.  Find,  to  five  significant  figures,  the  root  of  the  equation 

x^  +  2.73  x^  =  0.375. 

4.  Find  the  coordinates  of  the  intersections  of  the  curve  y 
=  {x-  l)2(x+2)  with  the  lines :  (a)  y  =  S;  (6)  y=l  x+1;  (c)  y=}x-l. 

5.  After  cutting  off  slices  of  thickness  1  in.,  1  in.,  2  in.,  parallel  to 
three  perpendicular  faces  of  a  cube,  the  volume  is  8  cu.  in.  What  was 
the  length  of  an  edge  of  the  cube  ? 

6.  Find  the  radius  of  that  sphere  whose  volume  is  decreased  50% 
when  the  radius  is  decreased  2  ft. 

7.  For  what  values  of  k  will  the  lines  kx  +  y  +  2  =  0,  x -\-  ky  —  1  =  0, 
2x—  y  -{-  k  =  0  pass  through  a  common  point  ? 

8.  For  what  values  of  k  are  the  following  equations  satisfied  by  other 
values  of  x,  y,  z,  to  than  0,  0,  0,  0?  kx  +  2y-{-z  —  Sw  =  0,  2x  +ky  +  z 
—  w  =  0,  X— 2y-^kz-{-w  =  0,  x+7y  —  z  +  kw=0- 

9.  A  buoy  composed  of  a  cone  of  altitude  6  ft.  surmounted  by  a 
hemisphere  with  the  same  base  when  submerged  displaces  a  volume  of 
water  equal  to  a  sphere  of  radius  6  ft.    Find  the  radius  of  the  buoy. 

10.  Find,  to  four  significant  figures,  the  coordinates  of  the  intersections 
of  the  parabolas  y -{- x^  =  7,  x  +  y^  =  U,  Ex.  13,  p.  138. 

11.  By  applying  Newton's  method  (§  164)  to  both  coordinate  axes 
simultaneously,  find  that  intersection  of  the  parabolas  x^  —  y  =  4  and 
X  +  y2  —  3  which  lies  in  the  first  quadrant. 

12.  The  segment  cut  out  of  a  sphere  of  radius  a  by  a  plane  through 
its  center  and  a  parallel  plane  at  the  distance  x  from  it  has  a  volume 
=  irx(a:^  —  iaj2);  at  what  distance  from  its  base  must  a  hemisphere  be 
cut  by  a  plane  parallel  to  the  base  to  bisect  the  volume  of  the  hemisphere  ? 


168  PLANE  ANALYTIC  GEOMETRY    [VIH,  §  171 

171.  Expansion  of  f{x  +  h).  The  solution  of  numerical  equa- 
tions is  based  on  the  fundamental  fact  (§  167)  that  if  f(x)  is  a  poly- 
nomial, then  f(x\  f  K)  can  be  expressed  as  a  polynomial  of  the  same 
degree  in  A,  and  the  coefficients  Aq,  J.i  ,  •••  J.„  of  this  polynomial  can  be 
calculated.      Thus,  for 

f(x)  —a^p^  +  aix^  +  a-ix^  +  a^x  +  a^ 
we  have : 

/(i»i+  h)  =  «o  (xi  +  hy  +  ai  (xi  +  hy  +  aa  (xi  +  hy+  as  (xi  +  h)+a4 
=  «o^i*  +  <^i^i^  +  «2a;i^  +  asXi  +  a4 
+  (4  aoXi^  +  3  aiXi^  +  2  a^xi  +  as)^ 
+  (6  aoXi2  +  3  aiXi  +  aa)^"^ 
+  (4aoXi +  ai)/i3 

+  ao^*. 

Now  this  process  is  closely  connected  with  that  of  finding  the  successive 
derivatives  of  the  polynomial.    Thus  we  have  for 

f{x)  =  uqX'^  +  aix^  +  a20c^  +  asx  +  at 
the  derivatives : 

f'{x)  =  4  aoSc8  +  3  aix^  +  2a2X  +  as, 
f"lx)  =  12  aoic2  +  6  aix  +  2  a2, 
/'"(x)  =  24aoa;  +  6ai, 
/-(x)=24ao, 

all  higher  derivatives  being  zero.  If  in  these  expressions  we  put  x  =  xi 
and  then  multiply  them  respectively  by  1 ,  /i,  h^/2  ! ,  h^/S  ! ,  h*/4: ! ,  and 
add,  we  find  precisely  the  above  expression  for  /(xi  +  h);  hence  we  have: 

whenever /(x)  is  a  polynomial  of  degree  4. 

It  can  be  proved  in  the  same  way  that  for  a  polynomial  of  degree  n 
we  have 

f(xi  +  h)=f(xO  +  f'ix^)h+^-^^h'^+--'+-^^^^^ 

This  formula  is  a  particular  case  of  a  general  proposition  of  the  differ- 
ential calculus,  known  as  Taylofs  theorem.  It  shows  that  the  value  of  a 
polynomial  for  any  value  x  =  Xi  -\-  h  can  he  found  if  we  know  the  value  of 
the  polynomial  itself  and  of  all  its  n  derivatives  for  some  particular 
value  xi  of  x.    This  property  is  characteristic  for  polynomials. 


CHAPTER   IX 


THE   PARABOLA 


172.  The  Parabola.  The  parabola  can  be  defined  as  the 
locus  of  a  point  whose  distance  from  a  fixed  point  is  equal  to  its 
distance  from  a  fixed  line.  The  fixed  point  is  called  the  focus, 
the  fixed  line  the  directrix,  of  the  parabola. 

Let  F  (Fig.  63)  be  the  fixed  point,  d  the  fixed  line ;  then 
every  point  P  of  the  parabola  must  satisfy 
the  condition 

FP=:PQ, 

Q  being  the  foot  of  the  perpendicular  from 
P  to  d.  Let  us  take  F  as  origin,  or  pole,  and 
the  perpendicular  FD  from  jP  to  the  directrix 
as  polar  axis,  and  let  the  given  distance  FD 
=  2  a.  Then  FP  =r  and  PQ  =  2  a —r  cos  cf>. 
The  condition  FP  =  PQ  becomes  therefore 

r  =  2  a  —  r  cos  cf>, 
2a 


I.e. 


(1) 


1  -f  cos  <|> 


This  equation,  which  expresses  the  radius  vector  of  P  as  a 
function  of  the  vectorial  angle  </>,  is  the  polar  equation  of  the 
parabola,  when  the  focus  is  taken  as  pole  and  the  perpendicular 
from  the  focus  to  the  directrix  as  polar  axis. 

173.  Polar  Construction  of  Parabolas.  By  means  of  the 
equation  (1)  the  parabola  can  be  plotted  by  points.  Thus,  for 
<^  =  0  we  find  r  =  a  as  intercept  on  the  polar  axis.  As  <^ 
increases  from  the  value  0,  r  continually  increases,  reaching 

169 


170 


PLANE  ANALYTIC   GEOMETRY       [IX,  §  173 


the   value  2  a  for   <^  =  i  tt,  and   becoming   infinite   as   <f>   ap- 
proaches the  vahie  tt. 

For  any  negative  value  of  <^  (between  0  and  —  tt)  the  radius 
vector  has  the  same  length  as  for  the  corresponding  positive 
value  of  <^ ;  this  means  that  the  parabola  is  symmetric  with 
respect  to  the  polar  axis. 

The  intersection  A  of  the  curve  with  its  axis  of  symmetry 
is  called  the  vertex,  and  the  axis  of 
symmetry  FA  the  axis,  of  the  parab- 
ola. The  segment  BB'  cut  off  by 
the  parabola  on  the  perpendicular  to 
the  axis  drawn  through  the  focus  is 
called  the  latus  rectum;  its  length 
is  4  a,  if  2  a  is  the  distance  between 
focus  and  directrix.  Notice  also  that 
the  vertex  A  bisects  this  distance 
FD  so  that  the  distance  between  focus 
and  vertex  as  well  as  that  between  vertex  and  directrix  is  a. 

In  Fig.  63  the  polar  axis  is  taken  positive  in  the  sense  from 
the  pole  toward  the  directrix.  If  the  sense  from  the  directrix 
to  the  pole  is  taken  as  positive  (Fig.  64),  we  have  again  with 
F  as  pole  FP  =  r,  but  the  distance  of  P  from  the  directrix  is 
2  a-\-r  cos  </>,  so  that  the  polar  equation  is  now 

(2)  ^=.— ^^- 

^  ^  1  — cos<^ 

We  have  assumed  a  as  a  positive  number,  2  a  denoting  the 
absolute  value  of  the  distance  between  the  fixed  point  (focus) 
and  the  fixed  line  (directrix).  The  radius  vector  r  is  then 
always  positive.  But  the  equations  (1)  and  (2)  still  represent 
parabolas  if  a  is  a  negative  number,  viz.  (1)  the  parabola  of 
Fig.  64,  (2)  the  parabola  of  Fig.  63,  the  radius  vector  r  being 
negative  (§  16), 


Fig.  G4 


IX,  §  175] 


THE  PARABOLA 


171 


Q 

llllllrm;;.^ 

4 

/ 

D 
d 

a\aF 

Fig.  65 


174.  Mechanical  Construction.  A  mechanism  for  tracing 
an  arc  of  a  parabola  consists  of  a  right- 
angled  triangle  (shaded  in  Fig.  65),  one  of 
whose  sides  is  applied  to  the  directrix. 
At  a  point  R  of  the  other  side  J?Q  a 
string  of  length  i^Q  is  attached ;  the  other 
end  of  the  string  is  attached  at  the  focus 
F.  As  the  triangle  slides  along  the  di- 
rectrix, the  string  is  kept  taut  by  means 
of  a  pencil  at  P  which  traces  the  parabola. 
Of  course,  only  a  portion  of  the  parabola  can  thus  be  traced, 
since  the  curve  extends  to  infinity. 

175.  Transformation  to  Cartesian  Coordinates.  To  obtain 
the  cartesian  equation  of  the  parabola  let  the  origin  0  be  taken 
at  the  vertex,  i.e.  midway  between  the  fixed  line  and  fixed 
point,  and  the  axis  Ox  along  the  axis  of  the  parabola,  positive 
in  the  sense  from  vertex  to  focus  (Fig.  Q>Q>).  Then  the  focus 
F  has  the  coordinates  a,  0,  and  the  equation  of  the  directrix  is 
X  =  —a.  The  distance  FP  of  any  point 
P{x,  y)  of  the  parabola  from  the  focus  is 
therefore  V(a;  —  ay  -f-  2/^  and  the  dis- 
tance QP  of  P  from  the  directrix  is 
a  +  x.     Hence  the  equation  is 

{x-ay  +  y^={a^x)\ 
which  reduces  at  once  to 
(3)  2/2  =  4  ax.  Fig.  66 

This  then  is  the  cartesian  equation  of  the  parabola,  referred 
to  vertex  and  axis,  I.e.  when  the  vertex  is  taken  as  origin  and 
the  axis  of  the  parabola  (from  vertex  toward  focus)  as  axis  Ox. 

Notice  that  the  ordinate  at  the  focus  (a,  0)  is  of  length  2  a ; 
the  double  ordinate  B'B  at  the  focus  is  the  latus  rectum  (§  173). 


172 


PLANE  ANALYTIC  GEOMETRY       [IX,  §  176 


176.  Negative  Values  of  a.  In  the  last  article  the  constant 
a  was  again  regarded  as  positive ;  but  (compare  §  173)  the  equa- 
tion (3)  still  represents  a  parabola  when  a  is  a  negative  number, 
the  only  difference  being  that  in  this  case  the  parabola  turns  its 
opening  in  the  negative  sense  of  the  axis  Ox  (toward  the  left 
in  Fig.  66).  Thus  the  parabolas  y^=4:ax  and  2/2=  —  4  ax  are  sym- 
metric to  each  other  with  respect  to  the  axis  Oy  (Ex.  14,  p.  138). 

The  equation  (3)  is  very  convenient  for  plotting  a  parabola 
by  points.  Sketch,  with  respect  to  the  same  axes,  the  parab- 
olas :  y^  =  16xj  y^  =  — 16  x,  y^  =  x,  y^=z  —  Xj  y'^=3xj  y'^=z  —  \  x, 

177.  Axis  Vertical.     The  equation 
(4)  x^^^ay, 

which  differs  from  (3)  merely  by  the  interchange  of  x  and  y, 
evidently  represents  a  parabola  whose  vertex  lies  at  the  origin 
and  whose  axis  coincides  with  the  axis  Oy.  The  parabolas  (3) 
and  (4)  are  each  the  reflection  of  the  other  in  the  line  y  =x 
(Ex.  14,  p.  138).     The  equation  (4)  can  be  written  in  the  form 

1 


y 


4a 


x\ 


As  1/4  a  may  be  any  constant,  this  is  the  equation  discussed  in 
§132. 

178.   New  Origin.     An  equation  of  the  form  (Fig.  67) 
(5)  (2/-/c)2  =  4a(a^-/0, 


__Vi^-. 


Q 
Fig.  68 


or  of  the  form  (Fig.  68) 

(6)  (x-hY  =  4.a{y-k\ 


IX,  §  179]  THE  PARABOLA  173 

evidently  represents  a  parabola  whose  vertex  is  the  point  (^,  k), 
while  the  axis  is  in  the  former  case  parallel  to  Ox,  in  the  latter 
to  Oy.  For,  by  taking  the  point  (/i,  k)  as  new  origin  we  can 
reduce  these  equations  to  the  forms  (3),  (4),  respectively. 

The  parabola  (5)  turns  its  opening  to  the  right  or  left,  the 
parabola  (6)  upward  or  downward,  according  as  4  a  is  positive 
or  negative. 

179.  General  Equation.  The  equations  (5),  (6)  as  well  as 
the  equations  (3),  (4)  are  of  the  second  degree.  Now  the 
general  equation  of  the  second  degree  (§  79), 

Ax^  +  2  Hxy  -^By^-\-2Gx-{-2Fy-\-C=0, 

can  be  reduced  to  one  of  the  forms  (5),  (6)  if  it  contains  no 
term  in  xy  and  only  one  of  the  terms  in  x"^  and  y^,  i.e.  if  H  =  0 
and  either  yl  or  jB  is  =0.  This  reduction  is  performed  (as  in 
§  80)  by  completing  the  square  my  ov  x  according  as  the  equa- 
tion contains  the  term  in  y"^  or  x"^. 

Thus  any  equation  of  the  second  degree,  containing  no  term  in 
xy  and  only  one  of  the  squares  x^,  y"^,  represents  a  parabola,  whose 
vertex  is  found  by  completing  the  square  and  whose  axis  is 
parallel  to  one  of  the  axes  of  coordinates. 

EXERCISES 

1.  Sketch  the  following  parabolas : 

(a)  r  = ? (6)  r  = — (c)  r  =  a  sec2  4  0. 

■^  1+COS0  ^    ^  1-COS0 

2.  Sketch  the  following  curves  and  find  their  intersections  : 

2  CL 

(a)  r  =  8  cos  <f>^  r  = (6)  r  =  a,  r  = 


1  —  cos  0  1  +  cos  0 

8  2  6E 

(c)  r  =  4  cos  0,  r  = (d)  r  cos<p  =  2  a,  r  = 

1  4-  cos  0        ^  ^  1  -  cos  0 

3.   Sketch  the  following  parabolas  : 
(a)  (y-2y  =  S(x^6).  (b)  (x  +  Sy  =  b(S  ^  y). 

(c)  x2  =  6(1/  +  1).  (d)  (y  +  3)2  =  -  3  X, 


174  PLANE  ANALYTIC  GEOMETRY       [IX,  §  179 

4.   Sketch  each  of  the  following  parabolas  and  find  the  coordinates  of 
the  vertex  and  focus,  and  the  equations  of  the  directrix  and  axis  : 
^  (a)  y^-2y-3x-2  =  0.  -  (6)  a;2  +  4  a;  -  4  ?/  =  0. 

(c)  x'^-Ax  +  Sy  +  l  =0.  (d)  Sx^-6x-  y  =  0. 

(c)  8y'^-16y-^x  +  6  =  0.  (/)  y2^y  +  x  =  0. 

(gr)  x2  -  X  -  3  ?/  +  4  =  0.  (/i)  8  ?/2  -  3x  +  3  =  0. 

6.    Sketch  the  following  loci  and  find  their  intersections  : 
'  («)  y  =  2x,  y  =  x2.  (&)  ?/^  =  4  ax,  x  +  ?/  =  3  a. 

(c)  y^  =  x  +  S,    2/2  =  6- X.  (d)  ?/2+4x+4=0,  x2  +  ?/2^41. 

6.  Sketch  the  parabolas  with  the  following  line^  and  points  as  direc- 
trices and  foci,  and  find  their  equations : 

—  (a)  X  -  4  =  0,  (6,  -  2).  (6)  y  +  3  =  0,   (0,  0). 
(c)  2x  +  5  =  0,   (0,   -1).                   (d)  x  =  0,  (2,   -3). 
(e)  3y-l=0,  (-2,  1).                  (/)  x  -  2  a  =  0,   (a,  b). 

7.  Find  the  parabola,  with  axis  parallel  to  Ox,  and  passing  through 
the  points : 

—  (a)   (1,0),  (5,4),  (10,  -6).  (&)  (-W,  -5),  (|,  0),  (|,  -3). 
(c)  (-1,  6),  (3,1),  (-V-,0). 

8.  Find  the  parabola,  with  axis  parallel  to  Ojy,  and  passing  through 
the  points : 

-•  (a)   (0,  0),  (-2,  1),  (6,  9).  (6)   (1,  4),  (4,  -  1),  (-3,  20). 

^(c)   (-2,1),  (2,  -7),  (-3,  -2). 

9.  Find  the  parabola  whose  directrix  is  the  line  3x  —  4y—  10  =  0 
and  whose  focus  is:  (a)  at  the  origin  ;  (b)  at  (5,  —  2).  Sketch  each  of 
these  parabolas.  When  does  the  equation  of  a  parabola  contain  an  xy 
term  ? 

10.  Find  the  parabolas  with  the  following  points  as  vertices  and  foci 
(two  solutions) : 

-(a)   (-3,  2),  (-3,  5).  (6)   (2,  5),  (-  1,  5). 

(c)  (-  1,  -  1),  (1,  -  1).  {d)   (0,  0),  (0,  -  a). 

11.  Show  that  the  area  of  a  triangle  whose  vertices  Pi  (xi,  yi), 
P2  {X2,  2/2),  P3  (X3 ,  2/3")  are  on  the  parabola  2/2  =  4  ax,  may  be  expressed 
by  the  determinant 

1/.  1 

=  ^(^2  -  2/3) (^3  -  yi){y2  -yi)' 
o  a 


1 

yi^  yi  1 

8a 

2/2^  2/2  I 

2/3^  2/3  1 

IX,  §  179]  THE  PARABOLA  175 

12.  The  area  J,  of  a  cross-section  of  a  sphere  of  radius  B,  at  a  distance 
h  from  the  surface,  is  given  by  the  formula 

A  =  2Ilh-h^  h<B. 

Reduce  this  equation  to  standard  form  A  =  kh^,  where  A  and  h  differ 
from  A  and  h  by  constants.     What  is  the  meaning  of  A  and  h  ? 

13.  Show  that  if  the  area  A  of  the  cross-section  of  any  solid  perpen- 
dicular to  a  line  Z,  at  a  distance  h  from  any  fixed  point  P  in  Z,  is  a  quad- 
ratic function  of  h  : 

A  =  ah'^  +  &A.  -t-  c  ; 

another  point  Q  in  I  exists,  such  that 

A  =  kh^, 
where  h  denotes  the  distance  from  Q  and  A  differs  from  J.  by  a  constant. 

14.  If  s  denotes  the  distance  (in  feet)  from  a  point  P  in  the  line 
of  motion  of  a  falling  body,  at  a  time  t  (in seconds), 

where  g  is  the  gravitational  constant  (32.2  approximately)  and  Sq  is  the 
distance  from  P  at  the  time  Iq,  show  that  this  equation  can  be  put  in 
the  standard  form 

s  =  hgT, 
where  s  denotes  the  distance  from  some  other  fixed  point  in  the  line  of 
motion  and  tis  the  time  since  the  body  was  at  that  point. 

15.  The  melting  point  t  (in  degrees  Centigrade)  of  an  alloy  of  lead  and 

zinc  is  found  to  be 

t  =  13S+ .S16x  + .01125  x% 

where  x  is  the  percentage  of  lead  in  the  alloy.     Reduce  the  equation  to 

standard  form  t  =  kx  \  and  show  that  x  —x  —  U^  t  =  t  —  k,  where  h  is 

the  percentage  of  lead  that  gives  the  lowest  melting  point,  and  k  is  the 

temperature  at  which  that  alloy  melts. 

16.  Show  that  the  locus  of  the  center  of  the  circle  which  passes 
through  a  fixed  point  and  is  tangent  to  a  fixed  line  is  a  parabola. 

17.  Show  that  the  locus  of  the  center  of  a  circle  which  is  tangent  to  a 
fixed  line  and  a  fixed  circle  is  a  parabola.  Find  the  directrix  of  this 
parabola. 

18.  Write  in  determinant  form  the  equation  of  the  parabola  through 
three  given  points,  Pi{xx,  ^/l),  Pi{x2,  1/2),  Psi^s,  2/3)  with  axis  parallel 
to  a  coordinate  axis. 


176 


PLANE  ANALYTIC  GEOMETRY       [IX,  §  180 


180.   Slope  of  the  Parabola.     The  slope  tan  a  of  the  parabola 

2/2  =  4  aa; 

at  any  point  P  (x,  y)  (Fig.  69)  can  be  found  (comp.  §  137)  by 
first  determining  the  slope 

tan «! = y^^y 

of  the  secant  PP^ ,  and  then  letting 

-f*i(^i?  Vi)  move  along  the  curve  up 

to  the  point  P(x,   y).     Now   as   Pj 

comes  to  coincide  with  P,  x^  becomes 

equal  to  x,  and  y^  equal  to  y,  so  that 

the  expression   for  tan  a^  loses  its 

meaning.    But  observing  that  P  and 

Pi  lie  on  the  parabola,  we  have  y"^  =  4  ax  and  y^ 

hence  y^  —  y'^  =  4ta(x^  —  x).      Substituting  from  this  relation 

the  value  of  x^  —  x  in  the  above  expression  for  tan  cti,  we  find 

for  the  slope  of  the  secant : 


4  axy^ ,  and 


tan  «i  =  4  a  -^ — —  = 

1  o  o 


4a 


2/i  -r     2/1  +  2/ 

If  we  now  let  Pj  come  to  coincidence  with  P  so  that  y^  becomes 
=  y,  we  find  for  the  slope  of  the  tangent  at  P(x,  y)  : 


(7) 


,             2a 
tana  = 

y 


This  slope  of  the  tangent  at  P  is  also  called  the  slope  of  the 
parabola  at  P.  The  ordinate  y  of  the  parabola  is  a  function  of 
the  abscissa  x ;  and  the  slope  of  the  parabola  at  P  (x,  y)  is  the 
rate  at  which  y  increases  with  increasing  £c  at  P;  in  other  words, 
it  is  the  derivative  y'  of  y  with  respect  to  x  (compare  §  138). 

As  by  the  equation  of  the  parabola  we  have  y  =  ±  2^ ax,  we 
find: 


IX,  §  182]  THE  PARABOLA  177 

(8)  2/'  =  tana  =  ?^=±J«. 

y         ^x 

The  double  sign  in  the  last  expression  corresponds  to  the  fact 
that  to  a  given  value  of  x  belong  two  points  of  the  curve  with 
equal  and  opposite  slopes. 

I     181.  Explicit  and  Implicit  Functions.    The  result  just  obtained 
that  when  2/2  =  4  ax  then  the  derivative  of  y  with  respect  to  x  is 

y 

can  he  derived  more  easily  by  the  general  method  of  the  differential  cal- 
culus.   This  requires,  however,  some  preliminary  explanations. 

In  the  cases  in  which  we  have  previously  determined  the  derivative  y' 
of  a  function  yotx  this  function  was  given  explicitly ;  i.e.  the  equation  be- 
tween X  and  y  that  represents  the  curve  was  given  solved  for  ?/,  in  the 
form  y=f(x). 

Our  present  equation  of  the  parabola,  ?/2  =  4  ax,  is  not  solved  for  y 
(though  it  could  readily  be  solved  for  y  by  writing  it  in  the  form 
y  =  ±  2y/ax)  ;  the  same  is  true  of  the  equation  of  the  circle  x^  +  y^  =  a^, 
or  more  generally  x^  +  y'^  +  ax  -\-  by  -{-  c  =  0,  and  also  of  the  general  equa- 
tion of  the  second  degree  (§79),  Ax^+2 Hxy  +  By^  +2  Gx +  2  Fy+G  =0. 
Such  equations  in  x  and  y,  whether  they  can  be  solved  for  y  or  not,  are 
said  to  give  y  implicitly  as  a  function  of  x.  For,  to  any  particular  value 
of  x  we  can  find  from  such  an  equation  the  corresponding  values  of  x 
(there  may  be  several  values  ;  and  they  may  be  real  or  imaginary).  Thus, 
any  equation  between  x  and  y,  of  whatever  form,  determines  y  as  a  func- 
tion of  X. 

182.  Derivatives  of  Implicit  Fimctions.  The  differential  cal- 
culus shows  that  to  find  the  derivative  ?/'  of  a  function  y  given  implicitly 
by  an  equation  between  x  and  y  we  have  only  to  differentiate  this  equation 
with  respect  to  x,  i.e.  to  find  the  derivative  of  each  term,  remembering 
that  y  is  a  function  of  x.  To  do  this  in  the  simple  cases  with  which  we 
shall  have  to  deal  we  need  only  the  following  two  propositions  {A)  and 
(5),  §§  183,  184. 

N 


178  PLANE  ANALYTIC   GEOMETRY       [IX,  §  183 

183.  (A)  Derivative  of  a  Function  of  a  Function,    if  u  is  a 

function  of  y,  and  y  a  function  of  x,  the  derivative  of  u  with  respect  to  x 
is  the  product  of  the  derivative  of  u  with  respect  to  y  into  the  derivative  y' 
of  y  with  respect  to  x. 

For,  as  u  is  a  function  of  y  which  itself  is  a  function  of  x,  u  is  also 
a  function  of  a;.  If  x  be  increased  by  Ax,  y  will  receive  an  increment  Ay 
and  u  an  increment  Aw.  We  want  to  find  the  derivative  of  u  with  respect 
to  X,  i.e.  the  limit  of  Au/Ax  as  Ax  approaches  zero.     Now  we  can  put 

Am  _  All    Ay  . 
Ax      Ay     Ax' 

the  limit  of  the  first  factor,  Au/Ay,  is  the  derivative  of  u  with  respect  to 
y,  while  the  limit  of  the  second  factor.  Ay/ Ax,  is  the  derivative  y'  of  y 
with  respect  to  x. 

Thus,  ii  u  =  y",  we  know  (§  151)  that  the  derivative  of  ii  with  respect 
to  y  is  =  ny'^~'^.  But  if  u  =  «/",  and  ?/  is  a  function  of  x,  we  can  also  find 
the  derivative  of  u  loith  respect  to  x;  by  the  proposition  {A)  it  is 
ny^~^  '  y' .  For  example,  suppose  that  ?«  =  ?/^,  where  i/=:x2  — .Sx,  so 
that  u  =  (x2  —  3  x)^.  Then  the  ^/-derivative  of  u  is  3  y'^ ;  but  the  x-de- 
rivative  of  m  is  3  ?/2  .  y'  =  3  y^{^  x  -  3)  =  3(x2  -  3  x)2(2  x  -  3).  This  can 
readily  be  verified  by  expanding  (x'-^  —  3x)3  and  differentiating  the  result- 
ing polynomial  in  the  usual  way  (§  150). 

184.  {B)  Derivative  of  a  Product,   if  n  and  v  are  functions  of  x, 

the  derivative  of  uv  is  u  times  the  derivative  of  v  plus  v  times  the  deriva- 
tive of  u  : 

derivative  of  uv  =  nv'  +  vu'. 

For,  putting  uv  =  y,  we  have  to  find  the  limit  of  Ay / Ax.  When  x  is  in- 
creased by  Ax,  u  receives  an  increment  Am,  v  an  increment  Av,  and  the 
increment  Ay  of  y  is  therefore 

Ay  =  {u  +  Au){v  +  Av)  —  uv  ; 
dividing  by  Ax,  we  find 

^  =  {n  ^  Au){y  ^  Av)  -  uv  ^^A?;_^^Am_^Am^^_ 

Ax  Ax  Ax         Ax      Ax, 

In  the  limit.  Ay / Ax  becomes  «/',  Av/Ax  becomes  u';  Auj A.x  becomes  u\ 
and  the  last  term  vanishes  because  its  factor  At?  becomes  zero.     Hence  : 
?/'  3=  uv'  +  vuK 


IX,  §  185]  THE  PARABOLA  179 

* 

185.   Computation  of  Derivatives  of    Lnplicit  Functions. 

We  are  now  prepared  to  find  the  derivative  of  y  when  y  is  given  im- 
plicitly as  a  function  of  x  by  the  equation  y'^  =  4  aa;.  We  have  only 
to  differentiate  this  equation  with  respect  to  a;,  i.e.  find  the  x-derivative 
of  each  term,  rememhering  that  y  is  a  function  of  x.  The  term  y^^  as 
a  function  of  a  function,  gives  2y  ■  y' ;  the  teiin  4 ax  gives  4 a  ;  hence 
we  find 

2  yy'  =z  4:  a,    whence      y'  =  — , 

y 

as  in  §  180. 

Similarly,  we  find  by  differentiating  the  equation  of  the  circle 

x2  +2/2=  a2 
that  2x+2yy'  =0, 

whence  y'  =  —  -; 

y 

i.e.  the  slope  of  the  circle  x^  +  y^  =  cfi  at  any  point  P(a;,  y)  is  minus  the 
reciprocal  of  the  slope  of  the  radius  through  P. 

If  y  is  given  implicitly  as  a  function  of  x  by  the  equation 

x2+  5x?/=  12, 

which,  as  we  shall  see  later,  represents  a  hyperbola,  we  find  the  derivative 
of  «/,  i.e.  the  slope  of  the  hyperbola,  by  differentiating  the  equation  and 
applying  to  the  second  term  the  proposition  {B)  : 

2x+6x-y'  +  ?/-5  =  0, 
whence  y/ ^  _  5  y  +  2  x  ^_  y  _  2 

5x      .        X      b 

EXERCISES 

1.  Find    the    derivative    of  u  with  respect  to  x  for  the  following 
functions : 

—  (a)  M  =  2/2,  when  1/ =3x  — 5,  (6)  u  =  y^-\-'^y.,Yf\iQr).y—x^—2x. 

-~{c)  M  =  2y3— 3?/2,when  y=x^-^x.        {d)  u=  ly^  —  y,  when  y  =  x^. 

2.  Find  the  slope  of  the  following  parabolas  at  the  point  P(x,  y)  : 
_(a)y^  =  bx.  (&)  2/2_5y  +  6x+4  =  0.  -(c)  3  2/2  =  4  x- 5. 

3.  Find  y'  for  the  following  products : 

^(a)  y  =x\x^  +  6x).  (6)  y  =  (x  +  S)(x- 6). 

(c)   y  =  (x-a)(x-b)(x-c).         (d)  y  =^  (x^  S)(2x  +  1). 


180  PLANE  ANALYTIC  GEOMETRY       [IX,  §  185 

4.  Find  the  slope  at  the  point  P(a-,  y)  for  each  of  the  following  circles 
by  differentiation  ;  compare  the  results  with  §§  88,  89  : 

(a)  x2  +  y2  =  12.  (6)  x^  +  y^  +  ax  +  by  +  c  =  0. 

(c)  Ax"^ -h  Ay^ -^  2  Gx -\- 2  Fy  +  C  =  0. 
\6.   Find -the  slope  y'  for  each   of  the  following  curves  at  the  point 
P(x,  y)  : 

(a)  xy  =  cCK  (b)  x^y  -  6x -\- 4  =  0. 

(c)  u4x2  +  2  Hxy  -{-  By^  +  2Gx-\-2Fy+  C  =  0. 

186.  Equation  of  the  Tangent.  As  the  slope  of  the 
parabola  f-  =  4.ax 

at  the  point  P{x,y)  is  2a/y  (§§180-185),  the  equation  of  the 
tangent  at  this  point  is 

F-2/  =  — (X-a;), 

y 

where  X,  Y  are  the  coordinates  of  any  point  of  the  tangent, 
while  Xj  y  are  the  coordinates  of  the  point  of  contact.     This 
equation  can  be  simplified   by  multiplying   both  sides  by  y 
and  observing  that  ?/^  =  4  ax ;  we  thus  find 
(9)  yY=2a{x+X). 

•Notice  that  (as  in  the  case  of  the  circle,  §  89)  the  equation 
of  the  tangent  is  obtained  from  the  equation  of  the  curve, 
7/2  =  4  ax,  by  replacing  y"^  hj  yY,2  xhj  x-{-  X. 

The  segment  TP  (Fig.  70)  of  the  tangent  from  its  intersec- 
tion T  with  the  axis  of  the 
parabola  to  the  point  of  contact 
P  is  called  the  leyigth  of  the 
tangent  at  P;  the  projection  TQ 
of  this  segment  TP  on  the  axis 
of   the    parabola   is   called   the  \ 

subtangent    at    P.     Now,    with  ^i^-  "^^ 

F=0,  equation  (9)  gives  X=—x,   i.e.   T0=  OQ;   hence  the 
subtangent  is  bisected  by  the  vertex.     This  furnishes  a  simple 


IX,  §  188]  THE  PARABOLA  181 

construction  for  the  tangent  at  any  point  P  of  the  parabola  if 
the  axis  and  vertex  of  the  parabola  are  known. 

187.  Equation  of  the  Normal.  The  normal  at  a  point  P 
of  any  plane  curve  is  defined  as  the  perpendicular  to  the  tan- 
gent through  the  point  of  contact. 

The  slope  of  the  normal  is  therefore  (§  27)  minus  the  recip- 
rocal of  that  of  the  tangent.  Hence  the  equation  of  the  normal 
to  the  parabola  is  : 

r-,=-A(x-.), 

that  is : 

(10)  yX-^2aY={2a-\-x)y. 

The  segment  PN  of  the  normal  from  the  point  P{x,  y) 
on  the  curve  to  the  intersection  N  of  the  normal  with  the  axis 
of  the  parabola  is  called  the  length  of  the  normal  at  P;  the 
projection  QN  oi  this  segment  P^on  the  axis  of  the  parabola 
is  called  the  subnormal  at  P. 

Now,  with  Y=  0,  equation  (10)  gives  X  =  2  a  -j-  a;,  and  as 
x=OQ,  it  follows  that  QN'—2a',  i.e.  the  subnormal  of  the 
parabola  is  constant,  viz.  equal  to  half  the  latus  rectum. 

188.  Intersections  of  a  Line  and  a  Parabola.  The  inter- 
sections of  the  parabola 

7/2  =  4  aa; 
with  the  straight  line 

y  =  mx  -f  b 

are  found  by  substituting  the  value  of  y  from  the  latter  in  the 

former  equation : 

(mx  -\-by  =  4:  ax, 
or,  reducing: 

m V  +  2  (m6  -  2  a)  i»  +  &^  =  0. 

The  roots  of   this  quadratic  in  x  are  the  abscissas  of  the 

points  of  intersection ;  the  ordinates  are  then  found  from 

y  =  mx  -f-  b. 


182  PLANE  ANALYTIC  GEOMETRY       [IX,  §  188 

It  thus  appears  that  a  straight  line  cannot  intersect  a  parabola 
in  more  than  two  points.  If  the  roots  are  imaginary,  the  line 
does  not  meet  the  parabola;  if  they  are  real  and  equal,  the 
line  has  but  one  point  in  common  with  the  parabola  and  is 
a  tangent  to  the  parabola  (provided  m  ^  0). 

189.  Slope  Equation  of  the  Tangent.     The  condition  for 

equal  roots  is 

(bm-2af  =  b'm\ 
which  reduces  to 

m  =  «. 
b 

The  point  that  the  line  of  this  slope  has  in  common  with  the 
parabola  is  then  found  to  have  the  coordinates 
2  a  —  bm      b^  ,  ,      o  i. 

m^  a 

As  the  slope  of  the  parabola  at  any  point  {x,  y)  is  (§  180) 
I/'  =  2  a/y,  the  slope  at  the  point  just  found  is  y'  =  a/b  =  m ; 
i.e.  the  slope  of  the  parabola  is  the  same  as  that  of  the  line 
y  =  mx-\-b;   this  line  is  therefore  a  tangent.     Thus,  the  line 

(11)  y  =  mx  H — 

m 

is  tangent  to  the  parabola  y^  =  4:  ax  whatever  the  value  of  m. 

This  may  be  called  the  slope-form  of  the  equation  of  the  tangent. 

Equation  (11)  can  also  be  deduced  from  the  equation  (9),  by 

putting  2  a/y  =  m  and  observing  that  2/^  =  4  ax. 

190.  Slope  Equation  of  the  Normal.  The  equation  (10)  of 
the  normal  can  be  written  in  the  form 

2a  2a 

or  since  by  the  equation  (3)  of  the  parabola  x  =  y^/A  a : 


Y=-JLx  +  y  +  -^' 
2a      ^^^Sa' 


IX,  §  191]  THE  PARABOLA  183 

If  we  denote  by  n  the  slope  of  this  normal,  we  have : 

71=-^,  y  =  -2an,  J--=-an\ 
2  a  Sa^ 

so  that  the  equation  of  the  normal  assumes  the  form 

(12)  r=  nX-2a7i-  an^. 

This  may  be  called  the  slope-form  of  the  equation  of  the  normal. 

191.  Tangents  from  an  Exterior  Point.  The  slope-form 
(11)  of  the  tangent  shows  that /rom  any  j^oint  (x,  y)  of  the  plane 
not  more  than  two  tangents  can  he  drawn  to  the  parabola  2/^  =  4  ax. 
For,  the  slopes  of  these  tangents  are  found  by  substituting  in 
(11)  for  X,  y  the  coordinates  of  the  given  point  and  solving  the 
resulting  quadratic  in  m.  This  quadratic  may  have  real  and 
different,  real  and  equal,  or  complex  roots. 

Those  points  of  the  plane  for  which  the  roots  are  real  and 
different  are  said  to  lie  outside  the  parabola ;  those  points  for 
which  the  roots  are  imaginary  are  said  to  lie  within  the  parab- 
ola; those  points  for  which  the  roots  are  equal  lie  on  the 
parabola. 

The  quadratic  in  m  can  be  written 

xm^  —  ym  -f-  a  =  0, 

so  that  the  discriminant  is  2/^  —  4  ax.  Therefore  a  point  (x,  y) 
of  the  plane  lies  within,  on,  or  outside  the  parabola  according  as 
y^  —  4:ax  is  less  than,  equal  to,  or  greater  than  zero. 

Similarly,  the  slope-form  (12)  of  the  normal  shows  that  not 
more  than  three  normals  can  be  drawn  from  any  point  of  the 
plane  to  the  parabola,  since  the  equation  (12)  is  a  cubic  for  n 
when  the  coordinates  of  any  point  of  the  plane  are  substituted 
for  X,  Y.  As  a  cubic  has  always  at  least  one  real  root  (§  160), 
there  always  exists  one  normal  through  a  given  point;  but 
there  may  be  two  or  three. 


184 


PLANE  ANALYTIC  GEOMETRY       [IX,  §  192 


192.  Geometric  Properties.  Let  the  tangent  and  normal 
at  P  (Fig.  71)  meet  the  axis  at  T,  N;  let  Q  be  the  foot  of  the 
perpendicular  from  P  to 
the  axis,  D  that  of  the  per- 
pendicular to  the  directrix 
d ;  and  let  0  be  the  vertex, 
F  the  focus. 

As  the  subtangent  TQ  is 
bisected  by  0  (§  186)  and 
J  the  subnormal  QN  is  equal 
to  2  a  (§  187),  while  0F= 
a,  it  follows  that  F  lies 
midway  between  T  and  TV. 

The  triangle  TPN  being  Fig.  71 

right-angled  at  P  and  F  being  the  midpoint  of  its  hypotenuse, 
it  follows  that  ^^p  _  ^rp_  ^-^ 

Hence,  if  axis  and  focus  are  given,  the  tangent  and  the  normal 
at  any  point  P  of  the  parabola  are  found  by  describing  about 
F  a  circle  through  P  which  will  meet  the  axis  at  T  and  N. 

As  FP=DP,  it  follows  that  FPDT  is  a  rhombus;  the 
diagonals  PT  and  FD  bisect  therefore  the  angles  of  the 
rhombus  and  intersect  at  right  angles.  As  TP  (like  TQ)  is 
bisected  by  the  tangent  at  the  vertex,  the  intersection  of  these 
diagonals  lies  on  this  tangent  at*  the  vertex.  The  properties 
just  proved  that  the  tangent  at  P  bisects  the  angle  between  the 
focal  radius  PF  and  the  parallel  PD  to  the  axis  and  that  the 
perpendicular  from  the  focus  to  the  tangent  meets  the  tangent  on 
the  tangent  at  the  vertex  are  of  particular  importance. 

193.  Diameters.  It  is  known  from  elementary  geometry  that 
in  a  circle  all  chords  parallel  to  any  given  direction  have  their 
midpoints  on  a  straight  line  which  is  a  diameter  of  the  circle. 


IX,  §  193] 


THE  PARABOLA 


185 


Similarly,  in  a  parabola,  the  locus  of  the  midpoiyits  of  all  chords 
parallel  to  any  given  direction  is  a  straight  line,  and  this  line 
which  is  parallel  to  the  axis 
is  called  a  diameter  of  the 
parabola.  To  prove  this,  take 
the  vertex  as  origin  and  the 
axis  of  the  parabola  as  axis  Ox 
(Fig.  72)  so  that  the  equation 
is  /  =  4  ax.  Any  line  of  given 
slope  m  has  the  equation 
y  =  mx  4-  6, 

and  with  variable  b  this  represents  a  pencil  of  parallel  lines. 
Eliminating  x  we  find  for  y  the  quadratic 


Fig.  72 


r 


i^2/  +  — =  0. 
m  m 


The  roots  ^i,  2/2  are  the  ordinates  of  the  points  P,,  P^  at 
which  the  line  intersects  the  parabola.     The  sum  of  the  roots  is 

4  a 

2/1  +  ^2  =  — ; 

m 
hence  the  ordinate  \{yi  +  2/2)  of  the  midpoint  P  between  P^ ,  P^ 
is  constant  (i.e.  independent  of  x),  viz.  =  2  a/m,  and  independ- 
ent of  b.  The  midpoints  of  all  chords  of  the  same  slope  m 
lie,  therefore,  on  a  parallel  to  the  axis,  at  the  distance  2  a/m 
from  it. 

The  condition  for  equal  roots  (§  189)  gives  b  =  a/m.  That 
one  of  the  parallels  which  passes  through  the  point  where  the 
diameter  meets  the  parabola  is,  therefore, 

,   a 
y  =  mx-^—] 
m 

by  §  189  this  is  a  tangent.     Thus,  the  tangent  at  the  end  of  a 
diameter  is  parallel  to  the  chords  bisected  by  the  diameter. 


[^ 


186  PLANE  ANALYTIC  GEOMETRY       [IX,  §  193 

EXERCISES 

1.  Find  and  sketch  the  tangent  and  normal  of  the  following  parabolas 
at  the  given  points  : 

(a)  2^2  =  25  X,  (2,  5).        (b)    Si/ =4x,  {S,  -  2).     (c)   y^  =  2x,{^,l). 
(d)  5y^=12x,{l-2).    (e)  i/^  =  x,{hl).  (/)  452/2  =  x,  (5,  1). 

2.  Show  that  the  secant  through  the  points  P(x,  y)  and  Pi  (xi ,  yi) 
of  the  parabola  i/2  =  4  ax  has  the  equation  4:aX—(y+yi)Y+yyi  =  0, 
and  that  this  reduces  to  the  tangent  at  P  when  Pi  and  P  coincide. 

3.  Find  the  angle  between  the  tangents  to  a  parabola  at  the  vertex 
and  at  the  end  of  the  latus  rectum.  Show  that  the  tangents  at  the  ends  of 
the  latus  rectum  are  at  right  angles. 

4.  Find  the  length  of  the  tangent,  subtangent,  normal,  and  subnormal 
of  the  parabola  y'^  z=4xa,t  the  point  (1,  2). 

5.  Find  and  sketch  the  tangents  to  the  parabola  y'^  =  Sx  from  each 
of  the  following  points  : 

(a)   (-  2,  3).  (b)  (-  2,  0).  (c)   (-  6,  0).  (d)   (8,  8). 

6.  Draw  the  tangents  to  the  parabola  y'^  =3x  that  are  inclined  to  the 
axis  Ox  at  the  angles :  («)  30°,  (6)  45^,  (c)  135°,  (d)  150° ;  and  find 
their  equations. 

7.  Find  and  sketch  the  tangents  to  the  parabola  ?/2  =  4  x  that  pass 
through  the  point  (—2,  2). 

8.  Find  and  sketch  the  normals  to  the  parabola  y^  =  6x  that  pass 
through  the  points  : 

(«)   (1,0).     (6)(V-, -3).     (c)(-V,  -f).     (^)(f,-|).     (e)   (0,0). 

9.  Are  the  following  points  inside,  outside,  or  on  the  parabola 
Sy^  =  x?     (a)  (3,1).     (5)  (2,  J),     (c)  (8,  |).     (c?)  (10,  f). 

10.  Show  that  any  tangent  to  a  parabola  intersects  the  directrix  and 
latus  rectum  (produced)  in  points  equally  distant  from  the  focus. 

11.  Show  that  the  tangents  drawn  to  a  parabola  from  any  point  of  the 
directrix  are  perpendicular. 

12.  Show  that  the  ordinate  of  the  intersection  of  any  two  tangents  to 
the  parabola  y^  =  i  ax  is  the  arithmetic  mean  of  the  ordlnates  of  the 
points  of  contact,  and  the  abscissa  is  the  geometric  mean  of  the  abscissas 
of  the  points  of  contact. 


IX,  §  193]        ^  THE  PARABOLA  187 

13.  Show  that  the  sum  of  the  slopes  of  any  two  tangents  of  the  parab- 
ola y^  =  4  ax  is  equal  to  the  slope  Y/Xof  the  radius  vector  of  the  point  of 
intersection  (X,  Y)  of  the  tangents  ;  find  the  product  of  the  slopes. 

14.  Find  the  locus  of  the  intersection  of  two  tangents  to  the  parabola 
2/2  =  4  ax,  if  the  sum  of  the  slopes  of  the  tangents  is  constant. 

-^ 15.    Find  the  locus  of  the  intersection  of  two  perpendicular  tangents  to 

a  parabola  ;  of  two  perpendicular  normals  to  a  parabola. 

16.  Show  that  the  angle  between  any  two  tangents  to  a  parabola  is 

half  the  angle  between  the  focal  radii  of  the  points  of  contact.  %vCy^\  li**'/^. 

17.  From  the  vertex  of  a  parabola  any  two  perpendicular  lines  are 
drawn  ;  show  that  the  line  joining  their  other  intersections  with  the 
parabola  cuts  the  axis  at  a  fixed  point. 

18.  Find  and  sketch  the  diameter  of  the  parabola  y^  =  6x  that  bisects 
the  chords  parallel  to  Sx  —  2y-\-5  =  0;  give  the  equation  of  the  focal 
chord  of  this  system. 

19.  Find  the  system  of  parallel  chords  of  the  parabola  y^  =  Sx  bisected 
by  the  line  y  =  S. 

20.  Find  the  diameter  and  corresponding  chord  of  the  parabola  y^=^x 
/      that  pass  through  the  point  (5,  —2)  ;  at  what  angle  does  this  diameter 

f*        meet  its  chord  ? 

21.  Show  that  the  tangents  at  the  extremities  of  any  chord  of  a  parab- 
ola intersect  on  the  diameter  bisecting  this  chord.     Compare  Ex.  12. 

22.  Find  the  length  of  the  focal  chord  of  a  parabola  of  given  slope  m. 

23.  Find  the  tangent  and  normal  to  the  parabola  x^  =  4  ay  in  terms  of 
the  coordinates  of  the  point  of  contact. 

24.  Find  the  angles  at  which  the  parabolas  y^  =  i.ax  and  x^  =  4ay 
intersect. 

25.  If  the  vertex  of  a  right  angle  moves  along  a  fixed  line  while  one 
side  of  the  angle  always  passes  through  a  fixed  point,  the  other  side 
envelopes  a  parabola  (i.e.  is  always  a  tangent  to  the  parabola) .  The  fixed 
line  is  the  tangent  at  the  vertex,  the  fixed  point  is  the  focus  of  the 
parabola. 

26.  Two  equal  confocal  parabolas  have  the  same  axis  but  open  in  op- 
posite sense  ;  show  that  they  intersect  at  right  angles. 


X       \rt 


188  PLANE  ANALYTIC  GEOMETRY       [IX,  §  193 

27.  If  axis,  vertex,  and  one  other  point  of  the  parabola  are  given,  ad- 
ditional points  can  be  constructed  as  follows  :  Let  O  be  the  vertex,  P  the 
given  point,  and  Q  the  foot  of  the  perpendicular  from  P  to  the  tangent 
at  the  vertex ;  divide  QF  into  equal  parts  by  the  points  A\,  ^2,  •••  ;  and 
OQ  into  the  same  number  of  equal  parts  by  the  points  By,  P2,  •••  ;  the 
intersections  of  O^i,  OA2,  •••  with  the  parallels  to  the  axis  through  Pi, 
P2,  •••  are  points  of  the  parabola. 

28.  If  two  tangents  AP^  AP2  to  a  parabola  with  their  points  of  con- 
tact Pi,  P2  are  given  and  ^Pi,  AP2  be  divided  into  the  same  number  of 
equal  parts,  the  points  of  division  being  numbered  from  Pi  to  A  and  from 
A  to  P2,  the  lines  joining  the  points  bearing  equal  numbers  are  tangents 
to  the  parabola.  To  prove  this  show  that  the  intersections  of  any  tangent 
with  the  lines  ^Pi,  ^P2  divide  the  segments  Pi^,  J.P2  in  the  same 
division  ratio. 

29.  The  shape  assumed  by  a  uniform  chain  or  cable  suspended  between 
two  fixed  points  Pi,  P2  is  called  a  catenary  ;  its  equation  is  not  algebraic 
and  cannot  be  given  here.  But  when  the  line  P1P2  is  nearly  horizontal 
and  the  depth  of  the  lowest  point  below  P1P2  is  small  in  comparison  with 
P1P2,  the  catenary  agrees  very  nearly  with  a  parabola. 

The  distance  between  two  telegraph  poles  is  120  ft.  ;  P2  lies  2  ft.  above 
the  level  of  Pi ;  and  the  lowest  point  of  the  wire  is  at  1/3  the  distance  be- 
tween the  poles.  Find  the  equation  of  the  parabola  referred  to  Pi  as 
origin  and  the  horizontal  line  through  Pi  as  axis  Ox  ;  determine  the  posi- 
tion of  the  lowest  point  and  the  ordinates  at  intervals  of  20  ft. 

30.  The  cable  of  a  suspension  bridge  assumes  the  shape  of  a  parabola 
if  the  weight  of  the  suspended  roadbed  (together  with  that  of  the  cables) 
is  uniformly  distributed  horizontally.  Suppose  the  towers  of  a  bridge 
240  ft.  long  are  60  ft.  high  and  the  lowest  point  of  the  cables  is  20  ft.  above 
the  roadway  ;  find  the  vertical  distances  from  the  roadway  to  the  cables 
at  intervals  of  20  ft. 

31.  When  a  parabola  revolves  about  its  axis,  it  generates  a  surface  called 
a  paraboloid  of  revolution ;  all  meridian  sections  (sections  through  the 
axis)  are  equal  parabolas.  If  the  mirror  of  a  reflecting  telescope  is  such 
a  surface  (the  portion  about  the  vertex) ,  all  rays  of  light  falling  in  parallel 
to  the  axis  are  reflected  to  the  same  point ;  explain  why. 


IX,  §  195]  THE  PARABOLA  189 

194.  Parameter  Equations.  Instead  of  using  the  cartesian 
or  polar  equation  of  a  curve  it  is  often  more  convenient  to 
express  x  and  y  (or  r  and  <^)  each  in  terms  of  a  third  variable, 
which  is  then  called  the  parameter. 

Thus  the  parameter  equations  of  a  circle  of  radius  a  about  the 
origin  as  center  are : 

x  =  a  cos  </),     y  =  a  sin  <j>, 
<f)  being  the  parameter.     To  every  value  of  <^  corresponds  a 
definite  x  and  a  definite  y,  and  hence  a  point  of  the  curve. 
The  elimination  of  <f),  by  squaring  and  adding  the  equations, 
gives  the  cartesian  equation  x^-^y^  =  o^. 

Again,  to  determine  the  motion  of  a  projectile  we  may  observe 
that,  if  gravity  were  not  acting,  the  projectile,  started  with  an 
initial  velocity  v^  at  an  angle  c  to  the  horizon  would  have  at  the 
time  t  the  position 

a;  =  Vo  cos  €  •  ^,     ?/  =  -Vo  sin  c  •  t, 
the  horizontal  as  well  as  the  vertical  motion  being  uniform. 
But,  owing  to  the  constant  acceleration  g  of  gravity  (down- 
ward), the  ordinate  y  is  diminished  by  ^gt"^  in  the  time  tj  so 
that  the  coordinates  of  the  projectile  at  the  time  t  are 

x  =  Vi)  cos  c  •  ^,     y  —  VQ^mc  't  —  ^  gt\ 
These  are  the  parameter  equations  of  the  path,  the  parameter 
here  being  the  time  t.    ■  The  elimination  of  t  gives  the  cartesian 
equation  of  the  parabola  described  by  the  projectile : 

y  =  Vota.n€'X-       J^        x\ 
2  Vq  cos^  c 

195.  Parameter  Equations  of  a  Parabola.    For  any  parabola 

2/2  =  4  dec  we  can  also  use  as  parameter  the  angle  a  made  by  the 

tangent  with  the  axis  Ox-,  we  have  for  this  angle  (§  180)  : 

,  2a 

tana  =  — ; 

y 

it  follows  that  y  =  2a  ctn  a  and  hence  x  =  y'^/A:  a=  a  ctn^  a. 


190  PLANE  ANALYTIC  GEOMETRY       [IX,  §  195 

The  equations 

X  —  a  ctn^  a,     y  =  2  a  ctn  a 

are  paramenter  equations  of  the  parabola  y^  =  4:ax;  the  elimina- 
tion of  cot  a  gives  the  cartesian  equation. 

196.  Parabola    referred  to  Diameter  and  Tangent.    The 

equation  of  the  parabola  y^  =  4iax  preserves  this  simple  form  if  instead  of 
axis  and  tangent  at  the  vertex  we  take  as 
axes  any  diameter  and  the  tangent  at  its  end. 
The  equation  in  these  oblique  coordinates  is 

yi^  =  4  aixi , 

where  ai  =  a/sin"^  a,  a  being  the  angle  betvi^een 
the  axes,  i.e.  the  slope  angle  at  the  tiq-w  origin 
Oi  (Fig.  73). 

To  prove  this  observe  that  as  the  new  origin 
0\_  {h,  k)  is  a  point  of  the  parabola  i/2  =  4  ax 
we  have  by  §  195 

h  =  a  ctn*  a,  k  =  2  a  ctn  a, 


y 

/r' 

^v 

*/ 

y^ 

/ 

/ 

u 

y 

/" 

h\ 

/  \ 

X 

/" 

/ 

V 

> 

Fig.  73 


a  being  the  angle  at  which  the  tangent  at  Oi  is  inclined  to  the  axis. 
Hence,  transferring  to  parallel  axes  through  Oi,  we  obtain  the  equation 


which  reduces  to 


+  2  a  ctn  ay  =  4  a  (x  +  a  ctn^  «), 


+  4  a  ctn  cc  .  1/  =  4  ax. 


The  relation  between  the  rectangular  coordinates  x,  y  and  the  oblique 
coordinates  Xi ,  yi ,  both  with  Oi  as  origin,  is  seen  from  the  figure  to  be 

X  =  xi  +  yx  cos  a,  y  =  yi  sin  a. 

Substituting  these  values  we  find 

yi^  sin2  ct  +  4  a  cos  « •  ?/i  =  4  axi  +  4  a!/i  cos  a. 


I.e. 


2/1^  =  4 


a 


sin  2  a 


xi  =  4  a\X\ 


if  we  put  a/sin2  a  —  a\. 


IX,  §  198] 


THE  PARABOLA 


191 


The  meaning  of  the  constant  ai  appears  by  observing  that 
sin2  a         tan2  ^j 


ai  = 


ai  is  therefore  the  distance  of  the  new  origin  0\  from  the  directrix,  or 
what  amounts  to  the  same,  from  the  focus  F. 

197.  Area  of  Parabolic  Segment.    A  parabola,  together  with 

any  chord  perpendicular  to  its  axis,  bounds  an  area  OPV^  (shaded  in 

Fig.  74).     It  was  shown  by  Archimedes  (about 

250  B.C.)  that  this  area  is  two  thirds  the  area 

of  the  rectangle  PP'Q'Q  that  has  the  chord 

P'P  as  one  side  and  the  tangent  at  the  vertex 

as  opposite  side.  'Yig.  74 

This  rectangle  PP'Q'Q  is  often  called  (somewhat  improperly)  the  cir- 
cumscribed rectangle  so  that  the  result  can  be  expressed  briefly  by  saying 
that  the  area  of  the  parabola  is  2/S  of  that  of  the  circumscribed  rectangle. 

This  statement  is  of  course  equivalent  to  saying  that  the  (non-shaded) 
area  OQP  is  1/3  of  the  area  of  the  rectangle  OQPB.  In  this  form  the 
proposition  is  proved  in  the  next  article, 

198.  Area  by  Approximation  Process.  To  obtain  first  an  ap- 
proximate value  {A)  for  the  area  OQP  (Fig.  75)  we  may  subdivide  the 
area  into  rectangular  strips  of  equal  width, 
by  dividing  OQ  into,  say,  n  equal  parts 
and  drawing  the  ordinates  ?/i ,  y^,  •••?/«. 
If  the  width  of  these  strips  is  Aic  so  that 
0Q  =  nAx,  we  have  as  approximate  value 
of  the  area : 

{A)  =  Aa; .  ?/i  +  Ax .  ?/2  + 


Fig.  75 


+  Ax  .  yn. 

Now  yi  is  the  ordinate  corresponding  to  the  abscissa  Ax  ;  ?/2  corresponds 
to  the  abscissa  2  Ax,  etc.  ;  ?/„  corresponds  to  the  abscissa  wAx  =  OQ. 
Hence,  if  the  equation  of  the  curve  is  x^  =  4  ay.,  we  have  : 


?/l=:-L(Ax)2,      ?/2  =  -1  (2  AX)2, 

4a  4a 


4a 


(wAx)2. 


Substituting  these  values  we  find  : 


{A)  = 


(Ax)3 
4a 


(1+22  +  32+    ...   +  W2), 


192 


PLANE  ANALYTIC  GEOMETRY       [IX,  §  198 


By  Ex.  3  6,  p.  74, 

1  +  22+    ...    +yj2 


i  «(n  +  1)(2  n  4- 1)  =  1(2  n3  +  3  n^  +  n)  ; 


hence 


(^)  =  IM'(2n3  +  3w2  +  «) 

^4  O, 


(tiAx) 
24 


^Y2 +?  +  !). 

a    V        n     n^J 


Now  nAx  =  OQ  =  Xn^  the  abscissa  of  the  terminal  point  P,  whatever  the 
number  n  and  length  Ax  of  the  subdivisions.  Hence,  if  we  let  the  num- 
ber n  increase  indefinitely,  we  find  in  the  limit  the  exact  expression  A  for 
the  area  OQP: 


12a     3  "'4a     3 


XnV^ 


where  y„  =  Xn^/4  a  is  the  ordinate  of  the  terminal  point  P.     As  x^n  is 
the  area  of  the  rectangle  OQPE,  our  proposition  is  proved. 

The  integral  calculus  furnishes  a  far  more  simple  and  more  general 
method  for  finding  the  area  under  a  curve.  The  method  used  above 
happens  to  succeed  in  the  simple  case  of  the  parabola  because  we  can 
express  the  sum  1  +  2^  +  3^  +  •••  +  w^  in  a  simple  form. 

199.  Area  expressed  in  Terms  of  Ordinates.  The  area 
(shaded  in  Fig.  76)  between  the  parabola  x^  =  4  a?/,  the  axis  Ox,  and  the 
two  ordinates  2/1,^3,  whose  abscissas  differ  by  y 
2  Ax  is  evidently,  by  the  formula  of  §  198, 

^  =  _l-(x33-Xi3)  =  J-[(xi  +  2Ax)3-Xin 
12  a  12  a 

=  j^  (6  xi2  +  12  XiAx  +  8  (Ax)2). 

1^  Gi 


Fig.  76 


This  expression  can  be  given  a  remarkably 
simple  form  by  introducing  not  only  the  ordinates  y\  —  XiV4  a,  y%  — 
(xi  +  2  Ax) 2/4  a,  but  also  the  ordinate  yi  midway  between  yi  and  1/3, 
whose  abscissa  is  x\  +  Ax.    For  we  have  : 

2/1+4^/2+^3  =i^[^i'  +  4(xi  +  Ax)2  +(xi  +  2  Ax)2] 
4a 

=  J_r6xi2  +  I2.X1AX  +  8(Axj2]. 
4  a 


IX,  §  200] 


THE  PARABOLA 


193 


y 

y. 

P^ 

-^ 

— ^^ 

Xt 

h 

m. 

X 

0 

Ax  Ax 

Fia.  77 


We  find  therefore : 

^  =  |Ax(yi +  4^/2  +  ^3). 

This  formula  holds  not  only  when  the  vertex  of  the  parabola  is  at  the 
origin,  but  also  when  it  is  at  any  point 
(A,  A;) ,  provided  the  axis  of  the  parabola 
is  parallel  to  Oy. 

For  (Fig.  77),  to  find  the  area  under 
the  arc  F1P2P3  we  have  only  to  add  to 
the  doubly  shaded  area  the  simply  shaded 
rectangle  whose  area  is  2  kAx.  We  find 
therefore  for  the  whole  area : 

\  Ax{yi  +  4  «/2  +  ys)  +  2  A:Aa;  =  i  Ax(yi  +  4  ya  +  2/3  +  6  fc) 

=  1  Aa;  [(2/1  +  A;)  +  4  (^2  +  k)  +(^3  +  A;)], 
where  yi,y2,  2/3  are  the  ordinates  of  the  parabola  referred  to  its  vertex, 
and  hence  yi  -\-  k,  y2  +  k,  ys -\-  k  the  ordinates  for  the  origin  O. 

We  have  therefore  for  any  parabola  whose  axis  is  parallel  to  Oy  : 

A  =  l  Ax(yi  +  4?/2  +  2/3). 

200.  Approximation  to  any  Area.    Simpson's  Rule.    The 

last  formula  is  sometimes  used  to  find  an  approximate  value  for  the  area 

under  any  curve  (i.e.  the  area  bounded 

by  the  axis  Ox,  an  arc  AB  of  the  curve, 

and  the  ordinates  of  A  and  B,  Fig.  78) . 

This  method  is  particularly  convenient 

if  a  number  of  equidistant  ordinates 

of  the  curve  are  known,   or  can  be 

found  graphically. 

Let  Ax  be  the  distance  of  the  ordi- 
nates, and  let  2/1,^2,  ys  be  any  three 
consecutive  ordinates.  Then  the  doubly  shaded  portion  of  the  required 
area,  between  yi  and  1/3,  will  be  (if  Ax  is  sufficiently  small)  very  nearly 
equal  to  the  area  under  the  parabola  that  passes  through  Pi ,  P2 ,  P3  and 
has  its  axis  parallel  to  Oy.    This  parabolic  area  is  by  §  199 

=  ^Aa;(yi+4?/2  +2/3). 
The  whole  area  under  AB  is  a  sum  of  such  expressions.     This  method 
for  finding  an  approximate  expression  for  the  area  under  any  curve  is 
o 


Fig.  78 


194 


PLANE  ANALYTIC  GEOMETRY       [IX,  §  200 


known  as  Simpson's  rule  (Thomas  Simpson,  1743)  although  the  funda- 
mental idea  of  replacing  an  arc  of  the  curve  by  a  parabolic  arc  had  been 
suggested  previously  by  Newton. 


Qj    Ax  Q;    Ax  Qj 
Fig.  79 


201.  Area  of  any  Parabolic  Segment.  As  the  equation  of  a 
parabola  referred  to  any  diameter  and  the  tangent  at  its  end  has  exactly 
the  same  form  as  when  the  parabola  is  referred  to  its  axis  and  the  tan- 
gent at  the  vertex  (§  196)  it  can  easily  be  shown  that  the  area  of  any 
parabolic  segment  is  2/3  of  the  area  of  the 
circumscribed  parallelogram.  In  this 
statement  the  parabolic  segment  is  under- 
stood t0j.be  bounded  by  any  arc  of  the 
parabola  and  its  chord;  and  the  circum- 
scribed parallelogram  is  meant  to  have  for 
two  of  its  sides  the  chord  and  the  parallel 
tangent  while  the  other  two  sides  are 
parallels  to  the  axis  through  the  extremities  of  the  chord  (Fig.  79). 

With  the  aid  of  this  proposition  Simpson's  rule  can  be  proved  very 
simply.  For,  the  area  of  the  parabolic  segment  P1P3P2  (Fig.  79)  is  then 
equal  to  2/3  of  the  parallelogram  formed  by  the  chord  P1P2,  the  tangent 
at  P2,  and  the  ordinates  yi,  ys  (produced  if  necessary).  This  parallelo- 
gram has  a  height  =  2  Ax  and  a  base  =  MP-z  =  ?/2  —  i  (2/1  +  y^)  ;  hence 
the  area  of  P1P3P2  is 

=  §  Aa;  (2  2/2  -  yi  -  2/3)  =  i  Ar  [4  ?/2  -  2  (yi  +  2/3)]. 

To  find  the  whole  shaded  area  we  have  only  to  add  to  this  the  area  of 
the  trapezoid  ^i§3P3Pi  which  is 

=  Ax(yi-\-y3). 


Hence  A  =  QiQsPsP^Pi  =  i  Ax[4  y, 
=  I-  Ax(yi  -f  4  ^2  +  yz) . 


2(2/1  + 2/3) +3(?/i  4-2/3)] 


EXERCISES 

1.  Show  that  the  area  of  any  parabolic  segment  is  2/3  of  the  area 
of  the  circumscribed  parallelogram. 

2.  In  what  ratio  does  the  parabola  y'^  =  4ax  divide  the  area  of  the 

circle  (x  —  a)'^  +  y^  =  4:a^? 


IX,  §201]  THE  PARABOLA  195 

3.  Find  the  area  bounded  by  the  parabola  i/^  =  4  ax  and  a  line  of 
slope  m  through  the  focus. 

4.  By  a  method  similar  to  that  used  in  finding  the  area  of  a  parabola 
(§  198),  find  exactly  the  area  bounded  by  the  curve  y  =  0(fi,  the  axis  Ox, 
and  the  line  x  =  a.  Wliat  is  the  area  bounded  by  this  same  curve,  the 
axis  Ox,  and  the  lines  x  =  a,  x  =  b?  What  is  the  area  bounded  by  the 
curve  y  =  x^  +  c,  the  axis  Ox,  and  the  lines  x  =  a,  x  =  6  ? 

5.  Find  and  sketch  the  curve  whose  ordinates  represent  the  area 
bounded  by  :  (a)  the  line  ?/  =  |  x,  the  axis  Ox,  and  any  ordinate,  (&)  the 
parabola  y  =  ^  x^,  the  axis  Ox,  and  any  ordinate. 

6.  Let  Pi(xi,  yi),  P2(xi  +  Ax,  2/2),  P3(xi  +  2  Ax,  ys)  be  three  points  of 
a  curve.  Let  A  denote  the  sum  of  the  areas  of  the  two  trapezoids  formed 
by  the  chords  P1P2 ,  P2P3 ,  the  axis  Ox,  and  the  ordinates  yi,  y^,  ys-  Let 
B  denote  the  area  of  the  trapezoid  formed  by  any  line  through  P2,  the 
axis  Ox,  and  the  segments  cut  off  on  the  ordinates  yi,  ys.  Find  the 
approximation  to  the  area  under  the  curve  given  by  each  of  the  following 
formulas:  ^(iA  +  B),  1(2  A +  B),  l(A  +  2B).  Which  of  these  gives 
Simpson's  rule  ? 

7.  To  find  an  approximation  to  the  area  bounded  by  a  curve,  the  axis 
Ox,  and  two  ordinates,  divide  the  interval  into  any  even  number  of  strips 
of  equal  width  and  apply  Simpson's  rule  to  each  successive  pair.  Show 
that  the  result  found  is  :  the  sum  of  the  extreme  ordinates  plus  twice  the 
sum  of  the  other  odd  ordinates  plus  four  times  the  sum  of  the  even  ordi- 
nates, multiplied  by  one  third  the  distance  between  the  ordinates. 

8.  Find  an  approximation  to  the  areas  bounded  by  the  following 
curves  and  the  axis  Ox  (divide  the  interval  in  each  case  into  eight  or 
more  equal  parts)  : 

(a)  4y  =  16-  x\  (6)  ?/  =  (x  +  3)  (x  -  2)2.  (c)  y=x'^-  x*. 

9.  "The  cross-sections  in  square  feet  of  a  log  at  intervals  of  6  ft.  are 
3.25,  4.27,  5.34,  6.02,  6.83  ;  find  the  volume. 

10.  The  cross  sections  of  a  vessel  in  square  feet  measured  at  intervals 
of  3  ft.  are  0,  2250,  6800,  8000,  10200  ;  find  the  volume.  Allowing  one 
ton  for  each  35  cu.  ft.,  what  is  the  displacement  of  the  vessel  ? 

11.  The  half-widths  in  feet  of  a  launch's  deck  at  intervals  of  5  ft.  are 
0,  1.8,  2.6,  3.2,  3.3,  3.3,  2.7,  2.1,  1  ;  find  the  area. 


196 


PLANE  ANALYTIC  GEOMETRY       [IX,  §  202 


202.  Shearing  Force  and  Bending  Moment.  A  straight 
beam  AB  (Fig.  80),  of  length  I,  fixed  at  one  end  ^  in  a  horizontal  posi- 
tion and  loaded  uniformlj'  with  w  lb.  per  unit  of  length,  will  bend  under 
the  load.  At  any  point  P,  at  the  distance  x  from  A,  the  efEect  of  the 
load  to(Z  —  x)  that  rests  on  PB  is  ^  ^ 

twofold : 

(a)  If  the  beam  were  cut  at  P,     yy^ 
this  load,  which  is  equivalent  to  a     ^^ 

single  force  W  =  w{l  —  x)  applied    'f^:A  P  B 

at  the  midpoint  of  PP,  would  pull      '■'^  Fig.  80  W-wfl-x) 

the  portion  PB  vertically  down. 

This  force  which  tends  to  shear  off  the  beam  at  P  is  called  the  shearing 
force  F  at  P.  Adopting  the  convention  that  downward  forces  are  to  be 
regarded  as  positive,  we  have 

F=w(l-x). 
The  shearing  force  at  the  various 
points  of  AB  is  therefore  repre- 
sented by  the  ordinates  of  the 
straight  line  CB  (Fig.  81) . 

(6)  If  the  beam  were  hinged  at  P,  the  effect  of  the  load  110(1  —  x)  on  PB 
would  be  to  turn  it  about  P.  As  the  force  w{l  —  x)  can  be  regarded  as 
applied  at  the  midpoint  of  PP,  this  effect  at  P  is  represented  by  the 
bending  moment  m  =  -\w{1-  xy, 

the  minus  sign  arising  from  the  convention  of  regarding  a  moment  as 
positive  when  tending  to  turn  counterclockwise.  As  w{l  —  x)  turns 
clockwise  about  P,  the  moment  is  ^,^ 
negative.  The  curve  DB  repre- 
senting the  bending  moments 
(Fig.  82)  is  a  parabola. 

More  briefly  we  may  say  that 
the  single  force  F=w{l  —  x) 
applied  at  the  midpoint  of  PB 
is  equivalent  to  an  equal  force 


c 

wl 

P 

""""■^^-1      ' 

-A 

Fig.  81 

^B 

Fig.  82 


at  P,  the  shear  F=w{l--x),  together  with  the  couple  formed  by  -f-P 
at  the  midpoint  of  PB  and  —  P  at  P ;  the  moment  of  this  couple  is  the 
bending  moment  M  =  —  \w{l  —  xy. 


IX,  §2031  THE  PARABOLA  197 

203.  Relation  of  Bending  Moment  to  Shearing  Force.     For 

any  beam  AB,  fixed  at  one  or  both  ends  or  supported  freely  at  two  or 
more  points,  in  a  horizontal  position,  and  loaded  by  any  vertical  forces, 
the  shearing  force  at  any  point  P  is  defined  as  the  algebraic  sum  of  all  the 
forces  (including  the  reactions  of  the  supports)  on  one  side  of  P,  and  the 
bending  moment  at  P  as  the  algebraic  sum  of  the  moments  of  these  forces 
about  P. 

It  may  be  noted  that  if  the  shear  Pis  constant,  the  bending  moment  is 
a  linear  function  of  x  (i.e.  of  the  abscissa  of  P) ;  if  P  (as  in  §  202)  is  a 
linear  function  of  x,  M  is  a.  quadratic  function  ;  in  either  case  the  deriva- 
tive of  M  with  respect  to  x  is  equal  to  P : 

M'  =  F. 

It  follows  that  the  bending  moment  is  a  maximum  or  minimum  at  any 
point  where  the  shear  is  zero. 

EXERCISES 

Determine  P  and  M  as  functions  of  x  for  a  horizontal  beam  AB  of 
length  I  and  represent  Pand  ilf  graphically  : 

1.  "When  the  beam  is  fixed  at  one  end  A  (cantilever)  and  carries 
a  single  load  W  at  the  other  end  B. 

2.  When  the  beam  is  freely  supported  at  its  ends  A,  B  and  loaded : 
(a)  uniformly  with  w  lb.  per  unit  of  length  ;  (&)  with  a  single  load  W  at 
the  midpoint ;  (c)  with  a  single  load  W  at  the  distance  a  from  A.  De- 
termine first  the  reactions  at  A  and  B. 

3.  When  the  beam  is  supported  at  the  two  points  trisecting  it  and 
carries :  (a)  a  uniform  load  w  lb. /ft. ;  (&)  a  single  load  W  a.t  A  and  at  B. 

4.  When  the  beam  is  supported  at  its  ends  and  is  loaded:  (a)  with 
w  lb. /ft.  over  the  middle  third  ;  (6)  with  w  lb. /ft.  over  the  first  and  third 
thirds;  (c)  with  w  Ib./ft.  over  the  first  half  and  2 to  lb. /ft.  over  the 
second  half. 

5.  When  the  beam  is  fixed  at  A  and  carries  w  lb. /ft.  over  the  outer 
half. 


CHAPTER   X 


Br- 


.>^ 


FiAt 


ELLIPSE  AND   HYPERBOLA 

204.  Definition  of  the  Ellipse.  The  ellipse  may  be  defined 
as  the  locus  of  a  point  whose  distances  from  two  fixed  points  have 
a  constant  sum. 

If  F^ ,  F2  (Fig.  83    are  the  fixed  points,  which  are  called  the 
foci,  and  if  P  is  any  point  of  the 
ellipse,  the  condition  to  be  satisfied  ^" 

by  P  is 

F^P  +  F.P  =  2  a. 

The  ellipse  can  be  traced  mechan- 
ically by  attaching  at  F^,  F^  the 
ends  of  a  string  of  length  2  a .  and  Fig.  83 

keeping  the  string  taut  by  means  of  a  pencil.  It  is  obvious 
that  the  curve  will  be  symmetric  with  respect  to  the  line  i^ii^2> 
and  also  with  respect  to  the  perpendicular  bisector  of  F1F2. 
These  axes  of  symmetry  are  called  the  axes  of  the  ellipse ;  their 
intersection  0  is  called  the  center  of  the  ellipse. 

205.  Axes.  The  points  A^,  A2,  B„  B2  (Figs.  83  and  84) 
vrhere  the  ellipse  intersects  these  axes  are  called  vertices. 
The  distance  ^2  A  of  those  vertices 
that  lie  on  the  axis  containing  the 
foci  Fi,  F2  is  =  2  a,  the  length  of 
the  string.  For  when  the  point  P 
in  describing  the  ellipse  arrives  at 
Ai,  the  string  is  doubled  along 
Fi  Ai  so  that  Fig.  m 

198 


X,  §2061  ELLIPSE  AND  HYPERBOLA  199 

and  since,  by  symmetry,  A>F.2  =^  F^A^,  we  have 
^2^2  +  F^F^  +  i^i  A  =  A  A  =  2  a. 
The  distance  A2A1  =  2  a,  which  is  called  the  major  axis,  must 
evidently  be  not  less  than  the  distance  F2F1  between  the  foci, 
which  we  shall  denote  by  2  c. 

The  distance  B2B1  of  the  other  two  vertices  is  called  the 
minor  axis  and  will  be  denoted  by  2  b.     We  then  have 

for  when  P  arrives  at  Bi,  we  have  B^F^  =  BiFi=  a. 

206.  Equation  of  the  Ellipse.  If  we  take  the  center  0  as 
origin  and  the  axis  containing  the  foci  as  axis  Ox,  the  equation  of 
the  ellipse  is  readily  found  from  the  condition  FiP-\-F2P=  2  a, 
which  gives,  since  the  coordinates  of  the  foci  are  c,  0  and 

-  c,  0 : 

^/{x  -  cf  +  2/'  4-  V(i^-  +  c)2  +  ?y2  =  2  a. 
Squaring  both  members  we  have 

a;24.2/2_|_c2  4.  V(aj2  4.^2_^c'_2  ex)  {x'^-\-y'^^c^+2  ex)  =  2  a^; 
transferring  x'^-\-y'^-\-G^  to  the  right-hand  member  and  squaring 
again,  we  find 

i.e.  (a2-c2)  a;2  4.ay=aXa2-c2). 

Now  for  the  ellipse  (§  205)  a^-c^=h\     Hence,  dividing  both 

members  by  aW,  we  find 

as  the  cartesian  equation  of  the  ellipse  referred  to  its  axes. 

This  equation  shows  at  a  glance :  (a)  that  the  curve  is  sym- 
metric to  Ox  as  well  as  to  Oy ;  (b)  that  the  intercepts  on  the 
axes  Ox,  Oy  are  ±a,  and  ±b.  The  lengths  a,  b  are  called  the 
semi-axes. 


200  PLANE  ANALYTIC  GEOMETRY         [X,  §  206 

Solving  the  equation  for  y  we  find 


h 


(2)  2/  =  ±-Va2-ar', 

a 

which  shows  that  the  curve  does  not  extend  beyond  the  vertex 

A^  on  the  right,  nor  beyond  A2  on  the  left. 

If  a  and  h  (or,  what  amounts  to  the  same,  a  and  c)  are  given 

numerically,  we  can  calculate  from  (2)  the  ordi nates  of   as 

many  points  as  we  please.     If,  in  particular,  a  —  h  (and  hence 

c  =  0)  the  ellipse  reduces  to  a  circle. 

EXERCISES 

1.  Sketch  the  ellipse  of  semi-axes  a  =  4,  6  =  3,  by  marking  the  ver- 
tices, constructing  the  foci,  and  determining  a  few  points  of  the  curve 
from  the  property  FiP  +  F2P  =  2  a.  Write  down  the  equation  of  this 
ellipse,  referred  to  its  axes. 

2.  Sketch  the  ellipse  x'^/W  +  y^/9  =  1  by  drawing  the  circumscribed 
rectangle  and  finding  some  points  from  the  equation  solved  for  y. 

3.  Sketch  the  ellipses  :  (a)  x^+2y'^  =  l.     (6)  Sx^-hl2y^  =  5. 

(c)  8  a:2  -}-  3  y2  =  20.         (d)  x^  -^  20  y^  =  1. 

4.  If  in  equation  (1)  a  <  ft,  the  equation  represents  an  ellipse  whose 
foci  lie  on  Oy.     Sketch  the  ellipses  : 

(a)  ^-1-1^=:  1.  (6)  20 x2 -H  ?/2  =  1.  (c)  10 x2 -H  9 2/2  =  10. 

4      16 

5.  Find  the  equation  of  the  ellipse  referred  to  its  axes  when  the  foci 
are  midpoints  between  the  center  and  vertices. 

6.  Find  the  product  of  the  slopes  of  chords  joining  any  point  of  an 
ellipse  to  the  ends  of  the  major  axis.  What  value  does  this  product 
assume  when  the  ellipse  becomes  a  circle  ? 

7.  Derive  the  equation  of  the  ellipse  with  foci  at  (0,  c),  (0,  -c),  and 
major  axis  2  a. 

8.  Write  the  equations  of  the  following  ellipses  :  (a)  with  vertices 
at  (5,  0),  (-  5,  0),  (0,  4),  (0,  -  4)  ;  (b)  with  foci  at  (2,  0),  (-  2.  0), 
and  major  axis  6. 

9.  Find  the  equation  of  the  ellipse  with  foci  at(l,  1),  (—1,  — 1), 
and  major  axis  6,  and  sketch  the  curve. 


X,  §  208] 


ELLIPSE  AND  HYPERBOLA 


201 


207.   Definition  of  the  Hyperbola.     The  hyperbola  can  be 

defined  as  tJie  locus  of  a  point  whose  distances  from  two  fixed 

points  have  a  coyistant  difference. 

The  fixed  points  F^,  F^  are  again  called  the  foci;  if  2  a  is 

the  constant  difference,  every  point  P  of  the  hyperbola  must 

satisfy  the  condition 

F^P-FJ'=±2a. 

Notice  that  the  length  2  a  must  here  be  not  greater  than  the 
distance  F^F^  =  2  c  of  the  foci. 

The  curve  is  symmetric  to  the  line  FiF^  and  to  its  perpen- 
dicular bisector. 

A  mechanism  for  tracing  an  arc  of  a  hyperbola  consists  of 
a  straightedge  F^Q  (Fig.  85)  which  turns  about  one  of  the 
foci,   F2 ;   a   string,   of  length   F2Q  —  2a,  is  fastened  to   the 


"> 


Fig.  85 

straightedge  at  Q  and  with  its  other  end  to  the  other  focus, 
Fi.  As  the  straightedge  turns  about  F2,  the  string  is  kept 
taut  by  means  of  a  pencil  at  P  which  describes  the  hyperbolic 
arc.  Of  course  only  a  portion  of  the  hyperbola  can  be  traced 
in  this  manner. 

208.  Equation  of  the  Hyperbola.  If  the  line  F2F1  be  taken 
as  the  axis  Ox,  its  perpendicular  bisector  as  the  axis  Oy,  and  if 
F2F1  =  2  c,  the  condition  F^P-  F^P=  ±  2  a  becomes  (Fig.  86)  : 


V(x-\-cy-\-f-V(x^cy-hy'=±2a, 


202  PLANE  ANALYTIC  GEOMETRY 

Squaring  both  members  we  find 


[X,  §  208 


squaring  again  and  reducing  as  in  §  206,  we  find  exactly  the 
same  equation  as  in  §  206 : 


Fig.  86 
But  in  the  present  case  c  ^  a,  while  for  the  ellipse  we  had 
c  <  a.     We  put,  therefore,  for  the  hyperbola 

the  equation  then  reduces  to  the  form 

which  is  the  cartesian  equation  of  the  hyperbola  referred  to  its  axes. 

209.  Properties  of  the  Hjrperbola.  The  equation  (3)  shows 
at  once:  (a)  that  the  curve  is  symmetric  to  Ox  and  to  Oy; 
(b)  that  the  intercepts  on  the  axis  Ox  are  ±  a,  and  that  the 
curve  does  not  intersect  the  axis  Oy. 

The  line  F2F1  joining  the  foci  and  the  perpendicular  bisector 
of  F2F1  are  called  the  axes  of  the  hyperbola ;  the  intersection 
0  of  these  axes  of  symmetry  is  called  the  center. 

The  hyperbola  has  only  two  vertices,  viz.  the  intersections 
Ai ,  A2  with  the  axis  containing  the  foci. 


X,  §210]  ELLIPSE  AND  HYPERBOLA  203 

The  shape  of  the  hyperbola  is  quite  different  from  that  of 
the  ellipse.     Solving  the  equation  for  y  we  have 


(4)  2/=±-Va^-a^ 

which  shows  that  the  curve  extends  to  infinity  from  A^^  to  the 
right  and  from  A^  to  the  left,  but  has  no  real  points  between 
the  lines  x  =  a,  x  =  —  a. 

The  line  F2F1  containing  the  foci  is  called  the  transverse 
axis;  the  perpendicular  bisector  of  F2F1  is  called  the  conjugate 
axis.  The  lengths  a,  h  are  called  the  transverse  and  conjugate 
semi-cfiXes. 

In  the  particular  case  when  a=b,  the  equation  (3)  reduces  to 
a?  —  y^  =  a^, 
and  such  a  hyperbola  is  called  rectangular  or  equilateral 

210.  Asymptotes.  In  sketching  the  hyperbola  (3)  or  (4)  it 
is  best  to  draw  first  of  all  the  two  straight  lines 


i.e. 

(5)  2/=±^^, 

which  are  called  the  asymptotes  of  the  hyperbola. 

Comparing  with  equation  (4)  it  appears  that,  for  any  value 
of  X,  the  ordinates  of  the  hyperbola  (4)  are  always  (in  absolute 
value)  less  than  those  of  the  lines  (5) ;  but  the  difference 
becomes  less  as  x  increases,  approaching  zero  as  x  increases  in- 
definitely. 

Thus,  the  hyperbola  approaches  its  asymptotes  more  and 
more  closely,  the  farther  we  recede  from  the  center  on  either 
side,  without  ever  reaching  these  lines  at  any  finite  distance 
from  the  center. 


204  PLANE  ANALYTIC  GEOMETRY         [X,  §  210 

EXERCISES 

1.  Sketch  the  hyperbola  x'^/XQ  —  y-2/4  =  1,  after  drawing  the  asymp- 
totes, by  determining  a  few  points  from  the  equation  solved  for  y  ;  mark 
the  foci. 

2.  Sketch  the  rectangular  hyperbola  cc^  —  2/2  —  9,  Why  the  name 
rectangular  ? 

3.  With  respect  to  the  same  axes  draw  the  hyperbolas : 

(a)  20x2  _  2/2  =  12.  (6)  a;2  -  20  2/2  =  12.  (c)  x^  -  y"^  =  12. 

4.  The  equation  —  x2/a2  +  y'^/h'^  =  1  represents  a  hyperbola  whose 
foci  lie  on  the  axis  Oy.    Sketch  the  curves  : 

(a)  -3x2  +  42/2  =  24.     (^b)  x^- Sy^ +  1S  =  0.     (c)  ^2 - 2/^  +  16  =  0. 

6.   Sketch  to  the  same  axes  the  hyperbolas  : 

^_y2=l      ^_2/2=_i. 
9       ^  '     9       ^ 

Two  such  hyperbolas  having  the  same  asymptotes  are  called  conjugate. 

6.  What  happens  to  the  hyperbola  a;2/a2  _  2/2/52  =  1  as  a  varies  ?  as 
b  varies  ? 

7.  The  equation  a;2/a2  —  y^/b^  =  k  represents  a  family  of  similar 
hyperbolas  in  which  k  is  the  parameter.  What  happens  as  k  changes 
from  1  to  —  1  ?     What  members  of  this  family  are  conjugate  ? 

8.  Find  the  foci  of  the  hyperbolas  : 

(a)  9  x2  -  16  ^2  =  144.  (5)  3  a;2  _  y2  =  12. 

9.  Find  the  hyperbola  with  foci  (0,  3),  (0,  —  3)  and  transvei-se  axis  4. 

10.  Find  the  equation  of  the  hyperbola  referred  to  its  axes  when  the 
distance  between  the  vertices  is  one  half  the  distance  between  the  foci. 

11.  Find  the  distance  from  an  asymptote  to  a  focus  of  a  hyperbola. 

12.  Show  that  the  product  of  the  distances  from  any  point  of  a  hyper- 
bola to  its  asymptotes  is  constant.     . 

13.  Find  the  hyperbola  through  the  point  (1,  1)  with  asymptotes 

y  =  ±2x. 

14.  Find   the   equation   of   the   hyperbola  whose   foci   are    (1,    1), 
(—1,  —  1),  and  transverse  axis  2,  and  sketch  the  curve. 


X,  §  212]  ELLIPSE  AND  HYPERBOLA  205 

211.  Ellipse  as  Projection  of  Circle.  If  a  circle  be  turned 
about  a  diameter  A2Ai  =  2a  through  an  angle  c(<|-7r)  and 
then  projected  on  the  original  plane,  the  projection  is  an 
ellipse. 

For,  if  in  the  original  plane  we  take  the  center  0  as  origin 
and  OAi  as  axis  Ox  (Fig.  87),  the 
ordinate  QP  of  every  point  P  of 
the  projection  is  the  projection  of 
the  corresponding  ordinate  QP^  of 
the  circle;  i.e. 

QP  =  QPi  cos  £.  Fio.  87 

The  equation  of  the  projection  is  therefore  obtained  from  the 
equation 

'    x'^-\- 1/^  =  0^ 

of  the  circle  by  replacing  y  by  y/cos  c.     The  resulting  equation 

COS^c 

represents  an  ellipse  whose  semi-axes  are  a,  the  radius  of  the 
circle,  and  b  —  a  cos  e,  the  projection  of  this  radius. 

212.  Construction  of  Ellipse  from  Circle.  We  have  just 
seen  that,  if  a  >  &,  the  ellipse 

a'     ¥ 
can  be  obtained  from  its  circumscribed  circle  x^  +  y^  =  a'^hj  re- 
ducing all  the  ordinates  of  this  circle  in  the  ratio  b/a.     This 
also  appears  by  comparing  the  ordinates 


y  =  ±Wa'-x' 
a 


of  the  ellipse  with  the  ordinates  y  =  ±  Va^  —  x"^  of  the  circle. 


206 


PLANE  ANALYTIC   GEOMETRY         [X,  §  213 


But  the  same  ellipse  can  also  be  obtained  from  its  inscribed 
circle  x^-\-y'^=  W  by  increasing  each  abscissa  in  the  ratio  a/h, 
as  appears  at  once  by  solving  for  x. 

It  follows  that  when  the  semi-axes  a,  h  are  given,  points  of 
the  ellipse  can  be  constructed  by  drawing  concentric  circles  of 
radii  a,  h  and  a  pair  of  perpendicular  diameters  (Fig.  88) ;  if 

y 


any  radius  meets  the  circles  at  P^,  P^  ?  the  intersection  P  of 
the  parallels  through  P^ ,  P^,  to  the  diameters  is  a  point  of  the 
ellipse. 

213.  Tangent  to  Ellipse.  It  follows  from  §  211  that  if 
P  (x,  y)  is  any  point  of  the  ellipse  and  P^  that  point  of  the  cir- 
cumscribed circle  which  has  the  same  abscissa,  the  tangents  at 
P  to  the  ellipse  and  at  P^  to  the  circle  must  meet  at  a  point  T  on 
the  major  axis  (Fig.  89). 


For,  as  the  circle  is  turned  about  A^Ai  into  the  position  in 
which  P  is  the  projection  of  Pj ,  the  tangent  to  the  circle  at  Pj 
is  turned  into  the  position  whose  projection  is  PT,  the  point  T 
on  the  axis  remaining  fixed. 


X,  §  214] 


ELLIPSE  AND  HYPERBOLA 


207 


The  tangent  XiX  +  yiY=  o?  to  the  circle  at  P^  [x^ ,  2/1)  meets 
the  axis  Ox  at  the  point  T  whose  abscissa  is 

Hence  the  equation  of  the  tangent  2XP{x,  y)  to  the  ellipse  is 

X     Y    1 
x     y     1 


0     1 


0, 


t.e. 


yX-fx--]Y-a^^  =  0; 
\        xj  X 


dividing  by  a^y/x  and  observing  that,  by  the  equation  of  the 
ellipse,  a;2  —  a^  =  —  (a'^/b'^)y'^  we  find 


(6) 


a2  ^  52 


as  equation  of  the  tangent  to  the  ellipse 

a""     b^ 
at  the  point  P(x,  y). 

-214.    Slope  of  Ellipse.     It  follows  from  the  equation  of  the 
tangent  that  the  slope  of  the  ellipse  at  any  point  P{x,  y)  is 

¥x 


tan  a  =  — 


a'y 


The  slope  being  the  derivative  y'  can  be  found  more  directly  by  differ- 
entiating the  equation  (1)  of  the  ellipse  (remembering  that  y  is  a  function 
of  X,  compare  §§  181-185) ;  this  gives 


whence 


2^  +  2^1^  =  0, 
a2        62 


?/'  =  tan  «=-  —  -. 


The  equation  (6)  of  the  tangent  is  readily  derived  from  this  value  of 
the  slope. 


208 


PLANE  ANALYTIC  GEOMETRY         [X,  §  215 


215.  Eccentricity.  For  the  length  of  the  focal  radius  F^P 
of  any  point  P(x,y)  of  the  ellipse  (1)  we  have  (Fig.  90), 
since  a^  —  6^  =  c^ : 

Fj^=(x-cy-{-y^={x-cy-\--^{a''-x')=\(a'--  2  a'^cx-^d'x''), 


whence 


F,P=± 


a x\ 

a  J 


The  ratio  c/a  of  the  distance  2  c  of  the  foci  to  the  major 
axis  2  a  is  called  the  (numerical) 
eccentricity   of  the    ellipse.     De- 
noting it  by  e  we  have 


FiP=±(a  —  ex)j 

and  similarly  we  find 

F,P=±(a  +  ex). 

For  the  hyperbola  (3)  we  find  in  the  same  way,  if  we  again 
put  e  =  c/a,  exactly  the  same  expressions  for  the  focal  radii 
F^P,  F2P(m  absolute  value).  Bat  as  for  the  ellipse  c^^a"^—  ¥ 
while  for  the  hyperbola  c^  =  a^-{-¥  it  follows  that  the  eccentrio- 
ity  of  the  ellipse  is  always  a  proper  fraction  becoming  zero  only 
for  a  circle,  while  the  eccentricity  of  the  hyperbola  is  always  greater 
than  one.  .  V 

216.  Equation  of  Normal  to  Ellipse.  As  the  normal  to  a 
curve  is  the  perpendicular  to  its  tangent  through  the  point  of 
contact,  the  equation  of  the  normal  to  the  ellipse  (1)  at  the  point 
P{x,  y)  is  readily  found  from  the  equation  (6)  of  the  tangent  as 


lX-^T=xy(^-^\  =  ^ 
¥         a"  \b^     ay     aW 


xy, 


I.e. 


«'x-^r=c^ 


X,  §  217] 


ELLIPSE  AND  HYPERBOLA 


209 


Tlie  intercept  made  by  this  normal  on  the  axis  Ox  is  there- 
fore 

ON=—x  =  e'^x. 


From  this  result  it  appears  by  §  215  that  (Fig.  91) 

F^N=  c  +  e^ic  =  e(a  -\-ex)=ze-  F,P, 
F^N=  c  -  e^a;  =  e(a  -ex)=e'  F^P-, 

hence  the  normal  divides  the  dis- 
tance F^Fi  in  the  ratio  of  the 
adjacent  sides  F2P,  F^P  of  the 
triangle  F.PF^.  It  follows  that 
the  normal  bisects  the  angle  between 
the  focal  radii  PFi ,  PF^ ;  in  other  words,  the  focal  radii  are 
equally  inclined  to  the  tangent. 

217.  Construction  of   any  H3rperbola   from  Rectangular 
Hyperbola.    The  ordinates  (4), 


Fig.  91 


y  =  ±--yx^—a\ 
a 

of  the  hyperbola  (3)  are  b/a  times  the  corresponding  ordinates 


y  =  ±  Va^  —  a^ 

of  the  equilateral  hyperbola  (end  of  §  209)  having  the  same 
transverse  axis.  When  6  <  a,  we  can  put  b/a  =  cos  €  and  re- 
gard the  general  hyperbola  as  the  projection  of  the  equilateral 
hyperbola  of  equal  transverse  axis.  When  6  >  a,  we  can  put 
a/b  =  cos  c  so  that  the  equilateral  hyperbola  can  be  regarded  as 
the  projection  of  the  general  hyperbola. 

In  either  case  it  is  clear  that  the  tangents  to  the  general  and 
equilateral  hyperbolas  at  corresponding  points  (i.e.  at  points 
having  the  same  abscissa)  must  intersect  on  the  axis  Ox. 


210  PLANE  ANALYTIC  GEOMETRY         [X,  §  218 

218.  Slope  of  Equilateral  Hyperbola.     To  find  the  slope  of 
the  equilateral  hyperbola 

x'2  -  y^  =  a\ 

observe  that  the  slope  of  any  secant  joining  the  point  P(x,y) 
and  Fi{xi,  y^)  is  {y^  —  y)/{x^—x),  and  that  the  relations 

y''=x^-a?, 
yi^  =  Xi^-a^ 

give       f-  -  y,^  =  X''  -  x,\  i.e.  (y  -  y,)(y  +  y,)  =(x-x,)(x  +  x^), 

whence  l^Uh^xJ^^ 

x-xi     y  +  yi 

Hence,  in  the  limit  when  P^  comes  to  coincidence  with  P,  we 
find  for  the  slope  of  the  tangent  at  P(x,  y)  : 

tan  a  =  ~' 

y 

The  equation  of  the  tangent  to  the  equilateral  hyperbola  is 
therefore 

y 

i.e.  since  x^  —y'^  =  a?: 

xX-yY=a\ 

219.  Tangent  to  the  Hyperbola.     It  follows  as  in  §  213  that 

the  tangent  to  the  geyieral  hyperbola  (3)  has  the  equation 

(7)  ^-^=1. 

The  slope  of  the  hyperbola  (3)  is  therefore 

y^x 


tan  a  = 


a^y 


This  slope  might  of  course  have  been  obtained  directly  by  differen- 
tiating the  equation  (3)  (compare  §  214). 


X,  §219]  ELLIPSE  AND  HYPERBOLA  211 

Notice  that  the  equations  (6),  (7)  of  the  tangents  are  obtained 
from  the  equations  (1),  (3)  of  the  curves  by  replacing  aj^,  ip-  by 
xX^  yY,  respectively  (compare  §§  89,  186). 

It  is  readily  shown  (compare  §  216)  that  for  the  hyperbola 
(3)  the  tangent  meets  the  axis  Ox  at  the  point  T  that  divides 
the  distance  of  the  foci  F^F^  proportionally  to  the  focal  radii 
F^P,  FiP,  so  that  the  tangent  to  the  hyperbola  bisects  the  angle 
between  the  focal  radii. 

EXERCISES      \  U 

1.  Show  that  a  right  cylinder  whose  cross-section  (i.e.  section  at 
right  angles  to  the  generators)  is  an  ellipse  of  semi-axes  a,  b  has  two 
(oblique)  circular  sections  of  radius  a  ;  find  their  inclinations  to  the 
cross-section. 

2.  Derive  the  equation  of  the  normal  to  the  hyperbola  (3) . 

3.  Find  the  polar  equations  of  the  ellipse  and  hyperbola,  with  the 
center  as  pole  and  the  major  (transverse)  axis  as  polar  axis. 

4.  Find  the  lengths  of  the  tangent,  subtangent,  normal,  and  sub- 
normal in  terms  of  the  coordinates  at  any  point  of  the  ellipse. 

5.  Show  that  an  ellipse  and  hyperbola  with  common  foci  are 
orthogonal. 

6.  Show  that  the  eccentricity  of  a  hyperbola  is  equal  to  the  secant 
of  half  the  angle  between  the  asymptotes. 

7.  Express  the  cosine  of  the  angle  between  the  asymptotes  of  a 
hyperbola  in  terms  of  its  eccentricity. 

8.  Show  that  the  tangents  at  the  vertices  of  a  hyperbola  intersect  the 
asymptotes  at  points  on  the  circle  about  the  center  through  the  foci. 

9.  Show  that  the  point  of  contact  of  a  tangent  to  a  hyperbola  is  the 
midpoint  between  its  intersections  with  the  asymptotes. 

10.  Show  that  the  area  of  the  triangle  formed  by  the  asymptotes  and 
any  tangent  to  a  hyperbola  is  constant. 

11.  Show  that  the  product  of  the  distances  from  the  center  of  a  hyper- 
bola to  the  intersections  of  any  tangent  with  the  asymptotes  is  constant. 

12.  Show  that  the  tangent  to  a  hyperbola  at  any  point  bisects  the  angle 
between  the  focal  radii  of  the  point.    [>^'l^  4  tUy  i^w^X-r^ 


/^   Z^^    ,,     jJHi^ 


212  PLANE  ANALYTIC   GEOMETRY         [X,  §  219 

13.  As  the  sum  of  the  focal  radii  of  every  point  of  an  ellipse  is  con- 
stant (§  204)  and  the  normal  bisects  the  angle  between  the  focal  radii 
(§  216),  a  sound  wave  issuing  from  one  focus  is  reflected  by  the  ellipse 
to  the  other  focus.  This  is  the  explanation  of  "  whispering  galleries." 
Find  the  semi-axes  of  an  elliptic  gallery  in  which  sound  is  reflected  from 
one  focus  to  the  other  at  a  distance  of  69  ft.  in  1/10  sec.  (the  velocity  of 

/        sound  is  1090  ft. /sec). 

14.  Show  that  the  distance  from  any  point  of  an  equilateral  hyperbola 
to  its  center  is  a  mean  proportional  to  the  focal  radii  of  the  point. 

15.  Show  that  the  bisector  of  the  angle  formed  by  joining  any  point 
of  an  equilateral  hyperbola  to  its  vertices  is  parallel  to  an  asymptote. 

16.  For  the  ellipse  obtained  by  turning  a  circle  of  radius  a  about  a 
diameter  through  an  angle  e  and  projecting  it  on  the  plane  of  the  circle, 
show  that  the  distance  between  the  foci  is  =  2  a  sin  e  ;  in  particular, 
show  that  the  foci  of  a  circle  are  at  the  center. 

17.  Show  that  the  tangents  at  the  extremities  of  any  diameter  (chord 
through  the  center)  of  an  ellipse  or  hyperbola  are  parallel. 

18.  Let  the  normal  at  any  point  Pof  an  ellipse  referred  to  its  axes  cut 
the  coordinate  axes  at  Q  and  B  ;  find  the  ratio  PQ/PB. 

19.  Show  that  a  tangent  at  any  point  of  the  circle  circumscribed  about 
an  ellipse  is  also  a  tangent  to  the  circle  with  center  at  a  focus  and  radius 
equal  to  the  focal  radius  of  the  corresponding  point  of  the  ellipse. 

20.  Show  that  the  lines  joining  any  point  of  an  ellipse  to  the  ends  of 
the  minor  axis  intersect  the  major  axis  (produced)  in  points  inverse  with 
respect  to  the  circumscribed  circle. 

21.  Show  that  the  product  of  the  ^/-intercept  of  the  tangent  at  any 
point  of  an  ellipse  and  the  ordinate  of  the  point  of  contact  is  constant. 

22.  Show  that  the  normals  to  an  ellipse  through  its  intersections  with 
a  circle  determined  by  a  given  point  of  the  minor  axis  and  the  foci  pass 
through  the  given  point. 

23.  Find  the  locus  of  the  center  of  a  circle  which  touches  two  fixed 
non-intersecting  circles. 

24.  Find  the  locus  of  a  point  at  which  two  sounds  emitted  at  an  inter- 
val of  one  second  at  two  points  2000  ft.  apart  are  heard  simultaneously. 


X,  §  222]  ELLIPSE  AND  HYPERBOLA  213 

220.  Intersections   of  a   Straight   Line  and  an  Ellipse. 

The  intersections  of  the  ellipse  (1)  with  any  straight  line  are 
found  by  solving  the  simultaneous  equations 

y  =  mx  -\-  k. 
Eliminating  y,  we  find  a  quadratic  equation  in  x : 

{w?a^  +  lf)x^  +  2  mka?x  +  {k^  -  h'^)a'  ==  0. 
To  each  of  the  two  roots  the  corresponding  value  of  y  results 
from  the  equation  y  =  mx  +  k. 

Thus,  a  straight  line  can  intersect  an  ellipse  in  not  more  than 
two  points. 

221.  Slope  Form  of  Tangent  Equations.  If  the  roots  of 
the  quadratic  equation  are  equal,  the  line  has  but  one  point  in 
common  with  the  ellipse  and  is  a  tangent. 

The  condition  for  equal  roots  is 

m'^k'^a^  =  (m^a""  +  b''){k''  -  b% 
whence  k  =  ±  Vm'^a^  -\-  ¥. 

The  two  parallel  lines 


(8)  y  =  mx±  Vm^a^  +  6^ 

are  therefore  tangents  to  the  ellipse  (1),  whatever  the  value  of 
m.  This  equation  is  called  the  slope  form  of  the  equation  of  a 
tangent  to  the  ellipse. 

It  can  be  shown  in  the  same  way  that  a  straight  line  cannot 
intersect  a  hyperbola  in  more  than  two  points,  and  that  the 
two  parallel  lines 

y  =  mx  ±  Vm^a^  —  b^ 

have  each  but  one  point  in  common  with  the  hyperbola  (3). 

222.  The  condition  that  a  line  be  a  tangent  to  an  ellipse  or 
hyperbola  assumes  a  simple  form  also  when  the  line  is  given 
in  the  general  form 

Ax-hBy-\-C=0. 


214  PLANE  ANALYTIC  GEOMETRY         [X,  §  222 

Substituting  the  value  of  y  obtained  from  this  equation  in 
the  equation  (1)  of  the  ellipse,  we  find  for  the  abscissas  of  the 
points  of  intersection  the  quadratic  equation : 

{A^o?  +  B^W)x'  H-  2  ACa^x  +  (C^  -B'¥)a^  =  0; 

the  condition  for  equal  roots  is 

which  reduces  to 

The  line  is  therefore  a  tangent  whenever  this  condition  is 
satisfied. 

When  the  line  is  given  in  the  normal  form, 

X  cos  p-\-ysm  p  =  p, 
the  condition  becomes 

p2  =  a2cos2;8-h62sin2^. 

223.  Tangents  from  an  Exterior  Point.    By  §  221  the  line 


y  =  mx  +  y/m'^a^  +  b^ 

is  tangent  to  the  ellipse  (1)  whatever  the  value  of  m.     The  condition  that 
this  hne  pass  through  any  given  point  (xi ,  yi)  is 


yi  =  mxi  +  Vm^a^  +  b^  ; 
transposing  the  term  mxi,  and  squaring,  we  find  the  following  quadratic 

equation  for  m  : 

to2xi2  -  2  mxiyi  +  yi^  =  mH^  +  6^ 

I.e.*  W  -  a^)w*^  -  2  ^i^iwi  +  y^  -  &2  =  0. 

The  roots  of  this  equation  are  the  slopes  of  those  lines  through  ix\ ,  y{) 

that  are  tangent  to  the  ellipse  (I). 

Thus,  not  more  than  two  tangents  can  be  drawn  to  an  ellipse  from  any 
point.  Moreover,  these  tangents  are  real  and  different,  real  and  coin- 
cident, or  imaginary,  according  as 


X,  §  225]  ELLIPSE  AND  HYPERBOLA  215 

This  condition  can  also  be  written  in  the  form 
6%i2  +  a^y{^  =  a2&2, 


I.e. 


Xi' 


Hence,  to  see  whether  real  tangents  can  be  drawn  from  a  point  (xi ,  yi) 
to  the  ellipse  (1)  we  have  only  to  substitute  the  coordinates  of  the  point 
for  X,  y  in  the  expression 

if  the  expression  is  zero,  the  point  (xi,  yi)  lies  on  the  ellipse,  and  only 
one  tangent  is  possible ;  if  the  expression  is  positive,  two  real  tangents 
can  be  drawn,  and  the  point  is  said  to  lie  outside  the  elHpse  ;  if  the  expres- 
sion is  negative,  no  real  tangents  exist,  and  the  point  is  said  to  lie  within 
the  ellipse. 

These  definitions  of  inside  and  outside  agree  with  what  we  would 
naturally  call  the  inside  or  outside  of  the  ellipse.  But  the  whole  discus- 
sion applies  equally  to  the  hyperbola  (3)  where  the  distinction  between 
inside  and  outside  is  not  so  obvious. 

224.  Symmetry.  Since  the  ellipse,  as  well  as  the  hyperbola, 
has  two  rectangular  axes  of  symmetry,  the  axes  of  the  curve, 
it  has  a  center,  the  intersection  of  these  axes,  i.e.  sl  point  of 
symmetry  such  that  every  chord  through  this  point  is  bisected 
at  this  point  (compare  §  135).  Analytically  this  means  that 
since  the  equation  (1),  as  well  as  (3),  is  not  changed  by  replac- 
ing a;  by  —  x,  nor  by  replacing  yhj—y,  it  is  not  changed  by 
replacing  both  x  and  y  by  —  x  and  —  ?/,  respectively.  In  other 
words,  if  {x,  y)  is  a  point  of  the  curve,  so  is  (—  a?,  —  y).  This 
fact  is  expressed  by  saying  that  the  origin  is  a  point  of  sym- 
metry, or  center. 

225.  Conjugate  Diameters.  Any  chord  through  the  center 
of  an  ellipse  or  hyperbola  is  called  a  diameter  of  the  curve. 


216 


PLANE  ANALYTIC  GEOMETRY         [X,  §  225 


Just  as  in  the  case  of  the  circle,  so  for  the  ellipse  the  locus 
of  the  midpoints  of  any  system  of  parallel  chords  is  a  diameter. 
This  follows  from  the  corresponding  property  of  the  circle 
because  the  ellipse  can  be  regarded  as  the  projection  of  a 
circle  (§211).  But  this  diameter  is  in  general  not  perpen- 
dicular to  the  parallel  chords ;  it  is  said  to  be  conjugate  to  the 
diameter  that  occurs  among  the  parallel  chords.  Thus,  in  Fig. 
92,  P'Q'  is  conjugate  to  PQ  (and  vice  versa). 


Fig.  92 

To  find  the  diameter  conjugate  to  a  given  diameter  y  =  mx. 
of  the  ellipse  (1),  let  y=mx-\-khe  any  parallel  to  the  given 
diameter.  If  this  parallel  intersects  the  ellipse  (1)  at  the  real 
points  (flJi,  ?/i)  and  (ajg,  2/2)?  t^ie  midpoint  has  the  coordinates 
^(xi  +  X2),  i(2/i  +  2/2)-  The  quadratic  equation  of  §  220  gives 
1  ,      ,      V  ma^k 

X  =  —  (X-,  -\- Xo)  = ^7 • 

If  instead  of  eliminating  y  we  eliminate  x,  we  obtain  the  quad- 
ratic equation 

(m^a^.+b^)y^  -  2  kh'y  +  (k^  -  m^a^)b^  =  0, 

whence 


1,      ,     .  b^k 


Eliminating  k  between  these  results,  we  find  the  equation  of  the 
locus  of  the  midpoints  of  the  parallel  chords  of  slope  m : 


X,  §  226]  ELLIPSE  AND  HYPERBOLA  217 

(9)  .  y  =  -^x. 

If  m  =  tan  a  is  the  slope  of  any  diameter  of  the  ellipse  (1), 
the  slope  of  the  conjugate  diameter  is 

mj  =  tan  cti  = -• 

ma^ 

The  diameter  conjugate  to  this  diameter  of  slope  m^  has  there- 
fore the  slope 

_        6^    _  ^'  _ 


\     mo?) 


i.e.  it  is  the  original  diameter  of  slope  m  (Fig.  92).  In  other 
words,  either  one  of  the  diameters  of  slopes  m  and  m^  is  conjugate 
to  the  other  ;  each  bisects  the  chords  parallel  to  the  other. 

226.  Tangents  Parallel  to  Diameters.  Among  the  parallel 
lines  of  slope  m,  y  =  mx  4-  Jc,  there  are  two  tangents  to  the 
ellipse,  viz.  (§  221)  those  for  which 


7c  =  ±  VmM  +  ^, 

their  points  of  contact  lie  on  (and  hence  determine)  the  conju- 
gate diameter.  This  is  obvious  geometrically;  it  is  readily 
verified  analytically  by.  showing  that  the  coordinates  of  the 
intersections  of  the  diameter  of  slope  —  li^/ma^  with  the 
ellipse  (1)  satisfy  the  equations  of  the  tangents  of  slope  m,  viz. 


y  =  mx  ±  ^m^a^  -f  6^. 

The  tangents  at  the  ends  of  the  diameter  of  slope  m  must  of 
course  be  parallel  to  the  diameter  of  slope  m-^.  The  four  tan- 
gents at  the  extremities  of  any  two  conjugate  diameters  thus 
form  a  circumscribed  parallelogram  (Fig.  92). 

The  diameter  conjugate  to  either  axis  of  the  ellipse  is  the 
other  axis  ;  the  parallelogram  in  this  case  becomes  a  rectangle. 


218 


PLANE  ANALYTIC   GEOMETRY         [X,  §  227 


227.  Diameters  of  a  Hyperbola.  For  the  hyperbola  the 
same  formulas  can  be  derived  except  that  ¥  is  replaced 
throughout  by  —  11^.  But  the  geometrical  interpretation  is 
somewhat  different  because  a  line  y  =  mx  meets  the  hyperbola 
(3)  in  real  points  only  when  m  <  b/a. 


Fig.  93 
The  solution  of  the  simultaneous  equations 
y  =  7nx, 
gives  : 


b'^x'^ 


ay  =  a^b^ 


x  =  ± 


ab 


V62 


y=± 


mob 


m^a^ 


Vb' 


7n^a^ 


These  values  are  real  if  m<b/a  and  imaginary  if  m>b/a 
(Fig.  93).  In  the  former  case  it  is  evidently  proper  to  call  the 
distance  PQ  between  the  real  points  of  intersection  a  diameter 
of  the  hyperbola  ;  its  length  is 

PQ  =  2  VS^+7^  =  2 «*  ^i^+MT. 

If  m>b/a,  this  quantity  is  imaginary;  but  it  is  customary  to 
speak  even  in  this  case  of  a  diameter,  its  length  being  defined 
as  the  real  quantity 

^  rn^a^  —  b^ 
By  this  convention  the  analogy  between  the  properties  of  the 
ellipse  and  hyperbola  is  preserved. 


X,  §  228]  ELLIPSE  AND   HYPERBOLA  219 

228.  Conjugate  Diameters  of  a  Hyperbola.  Two  diameters 
of  the  hyperbola  are  called  conjugate  if  their  slopes  7n,  mi  are 
such  that 

mrrii  =  — 

One  of  these  lines  evidently  meets  the  curve  in  real  points,  the 
other  does  not. 

If  m  <  b/a,  the  line  y  =  mx,  as  well  as  any  parallel  line, 
meets  the  hyperbola  (3)  in  two  real  points,  and  the  locus  of  the 
midpoints  of  the  chords  parallel  to  y  =  mx  is  found  to  be  the 
diameter  conjugate  to  y  —  mx,  viz. 

y  =  miX  =  — -  X. 
ma^ 

If  m  >  b/a,  the  coordinates  a^,  yi  and  ajg,  2/2  of  the  intersec- 
tions of  y=zmx  with  the  hyperbola  are  imaginary;  but  the 
arithmetic  means  ^  (X1  +  X2),  ^(?/i  +  ?/2)  ^i'^  real,  and  the  locus 
of  the  points  having  these  coordinates  is  the  real  line 

b' 
y  =  miX  =  —  X. 
ma^ 

It  may  finally  be  noted  that  what  was  in  §  227  defined  as 
the  length  of  a  diameter  that  does  not  meet  the  hyperbola 

in  real  points  is  the  length  of  the  real  diameter  of  the  hyper- 
bola 

•     -^  +  ^'  =  1; 
d?       b"" 

two  such  hyperbolas  are  called  conjugate. 


220 


PLANE  ANALYTIC  GEOMETRY         [X,  §  229 


229.   Parameter  Equations.     Eccentric  Angle.     Just  as  the 
parameter  equations  of  the  circle  x"^  -\-  y"^  =  o?  are  (§  194) : 

ic  =  a  cos  ^,  y  =  a  sin  0, 
so  those  of  the  ellipse  (1)  are 

ic  =  a  cos  dy  y=h  sin  d, 
and  those  of  the  hyperbola  (3)  are 

a:  =  a  sec  ^,  y  =h  tan  6. 
In  each  case  the  elimination  of  the  parameter  $  (by  squaring 
and  then  adding  or  subtracting)  leads  to  the  cartesian  equation. 

The  angle  6,  in  the  case  of  the 
circle,  is  simply  the  polar  angle  of 
the  point  P  (x,  y).  In  the  case  of  the 
ellipse,  as  appears  from  Fig.  94 
(compare  §  212),  6  is  the  polar  angle 
not  of  the  point  P  {x,  y)  of  the  ellipse, 
but  of  that  point  Pi  of  the  circum-r 
scribed  circle  which  has  the  same 
abscissa  as  P,  and  also  of  that  point 
Pg  of  the  inscribed  circle  which  has  the  same  ordinate  as  P. 
This  angle  6  =  xOP^  is  called  the  eccentric  angle  of  the  point 
P  (a;,  y)  of  the  ellipse. 

In  the  case  of  the  hyperbola  the  eccentric  angle  6  determines 
the  point  P(x,  y)  as  follows  (Fig.  95).  Let  a  line  through  0 
inclined  at  the  angle  6  to  the  trans- 
verse axis  meet  the  circle  of  radius 
a  about  the  center  at  A,  and  let  the 
transverse  axis  meet  the  circle  of 
radius  h  about  the  center  at  B.  Let 
the  tangent  at  A  meet  the  transverse 
axis  at  A'  and  the  tangent  at  B  meet 
the  line  OA  at  B'.  Then  the  parallels  to  the  axes  through^' 
and  B'  meet  at  P. 


Fig.  94 


Fig.  05 


X,  §  230]  ELLIPSE  AND  HYPERBOLA     *  221 

230.  Area  of  Ellipse.  Since  any  ellipse  of  semi-axes  a,  b 
can  be  regarded  as  the  projection  of  a  circle  of  radius  a, 
inclined  to  the  plane  of  the  ellipse  at  an  angle  €  such  that 
cos  €  =  b/a,  the  area  A  of  the  ellipse  is  ^  =  vd^  cos  c  =  -n-ab. 

EXERCISES 

1.   Find  the  tangents  to  the  ellipse  x^  +  iy^  =  16,  which  pass  through 
the  following  points : 

(a)  (2,  V3),    (b)  (-3,iV7),    (c)  (4,0),    (d)  (-8,0). 
\    2.    Find  the  tangents  to  the  hyperbola  2  x^  —  S  y^  =  IS,  which  pass 
through  the  following  points  : 

(a)  (-6,  3V2),    (&)   (-3,0),    (c)   (4,  -V5),    (d)  (0,0). 
,-^   3.   Find  the  intersections  of  the  line  x  —  2y  =  7  and  the  hyperbola 

x^-y^  =  5. 
4.   Find  the  intersections  of  the  line  Sx  +  y  —  1  =  0  and  the  ellipsQ 
x^  +  4y^  =  65. 
. ;  ^'5.  For  what  value  of  k  will  the  line  y  =  2x  +  khe  3,  tangent  to  the 
hyperbola  ic2-4y2-4  =  0? 

-^    6.  For  what  values  of  m  will  the  line  y=7nx  +  2  be  tangent  to  the 
ellipse  x2  +  4  2/2  _  1  =  0  ? 

7.  Find  the  conditions  that  the  following  lines  are  tangent  to  the  hy- 
perbola x2/a2  -  1/2/62  =  1  . 

(a)  Ax  +  By  -{-  C  =  0,    (b)  xcos^  +  y  sin p  =p. 

8.  Are  the  following  points  on,  outside,  or  inside  the  ellipse  ^2+4  y2=4p 

(«)  (1,1),    (b)  (I,  -i),    (c)  (-i,  -I). 

9.  Are   the  following    points  on,  outside,  or  inside   the   hyperbola 
9x2-2/2  =  9?  (^a)  (f,  -I),    (6)  (1.35,2.15),    (c)   (1.3,2.6). 

~^  10.   Find  the  difference  of  the  eccentric  angles  of  points  at  the  extremi- 
ties of  conjugate  diameters  of  an  ellipse. 

11.    Show  that  conjugate  diameters  of  an  equilateral  hyperbola  are 
equal. 
f-  12.  Show  that  an  asymptote  is  its  own  conjugate  diameter. 

-  13.   Show  that  the  segments  of  any  line  between  a  hyperbola  and  its 
asymptotes  are  equal. 

-  14.   Find  the  tangents  to  an  ellipse  referred  to  its  axes  which  have 
equal,  intercepts. 


222  PLANE  ANALYTIC   GEOMETRY         [X,  §  230 

15.  What  is  the  greatest  possible  number  of  normals  that  can  be  drawn 
from  a  given  point  to  an  ellipse  or  hyperbola  ? 

16.  Show  that  tangents  drawn  at  the  extremities  of  any  chord  of  an 
ellipse  (or  hyperbola)  intersect  on  the  diameter  conjugate  to  the  chord. 

17.  Show  that  lines  joining  the  extremities  of  tlie  axes  of  an  ellipse 
are  parallel  to  conjugate  diameters. 

18.  Show  that  chords  drawn  from  any  point  of  an  ellipse  to  the  ex- 
tremities of  a  diameter  are  parallel  to  conjugate  diameters. 

19.  Find  the  product  of  the  perpendiculars  let  fall  to  any  tangent  from 
the  foci  of  an  ellipse  (or  hyperbola). 

20.  The  earth's  orbit  is  an  ellipse  of  eccentricity  .01677  with  the  sun 
at  a  focus.  The  mean  distance  (major  semi-axis)  between  the  sun  and 
earth  is  93  million  miles.  Find  the  distance  from  the  sun  to  the  center 
of  the  orbit. 

21.  Find  the  sum  of  the  squares  of  any  two  conjugate  semi-diameters 
of  an  elUpse.  Find  the  difference  of  the  squares  of  conjugate  semi-diam- 
eters of  a  hyperbola. 

22.  Find  the  area  of  the  parallelogram  circumscribed  about  an  ellipse 
with  sides  parallel  to  any  two  conjugate  diameters. 

23.  Find  the  angle  between  conjugate  diameters  of  an  ellipse  in  terms 
of  the  semi-diameters  and  semi-axes. 

24.  Express  the  area  of  a  triangle  inscribed  in  an  ellipse  referred  to 
its  axes  in  terms  of  the  eccentric  angles  of  the  vertices. 

25.  The  circle  which  is  the  locus  of  the  intersection  of  two  perpendicu- 
lar tangents  to  an  ellipse  or  hyperbola  is  called  the  director-circle  of  the 
conic.     Find  its  equation  :  {a)  For  the  ellipse.     (&)  For  the  hyperbola. 

26.  Find  the  locus  of  a  point  such  that  the  product  of  its  distances 
from  the  asymptotes  of  a  hyperbola  is  constant.  For  what  value  of  this 
constant  is  the  locus  the  hyperbola  itself  ? 

27.  Find  the  locus  of  the  intersection  of  normals  drawn  at  correspond- 
ing points  of  an  ellipse  and  the  circumscribed  circle. 

28.  Two  points  J.,  J5  of  a  line  I  whose  distance  is  AB  =  a  move  along 
two  fixed  perpendicular  lines  ;  find  the  path  of  any  point  P  of  I. 


CHAPTER   XI 


CONIC   SECTIONS  — EQUATION   OF   SECOND   DEGREE 
PART   I.     DEFINITION   AND   CLASSIFICATION 

231.  Conic  Sections.  The  ellipse,  hyperbola,  and  parabola 
are  together  called  conic  sections,  or  simply  conies,  because 
the  curve  in  which  a  right  circular  cone  is  intersected  by  any 
plane  (not  passing  through  the  vertex)  is  an  ellipse  or  hyper- 
bola according  as  the  plane  cuts  only  one  of  the  half-cones  or 
both,  and  is  a  parabola  when  the  plane  is  parallel  to  a  gener- 
ator of  the  cone.  This  will  be  proved  and  more  fully  dis- 
cussed in  §§  239-243. 

232.  General  Definition.  The  three  conies  can  also  be 
defined  by  a  common  property  in  the  plane :  the  locus  of  a  point 
for  ivhich  the  ratio  of  its  distances  from  a  fixed  point  and  from 
a  fixed  line  is  constant  is  a  conic,  viz.  an  ellipse  if  the  constant 
ratio  is  less  than  one,  a  hyperbola  if 
the  ratio  is  greater  than  one,  and  a 
parabola  if  the  ratio  is  equal  to  one. 

We  shall  find  that  this  constant 
ratio  is  equal  to  the  eccentricity  e  —  cja 
as  defined  in  §  215.  Just  as  in  the 
case  of  the  parabola  for  which  the 
above  definition  agrees  with  that  of 
§  172,  we  shall  call  the  fixed  line  d^  directrix,  and  the  fixed 
point  jPj  focus  (Fig.  96). 

223 


y 

L 

^ 

__$ 

/f 

i 

D 

X 

Fi 

<---, 

K.- 

--^ 

iL 

y 

d, 

Fig.  96 


224  PLANE  ANALYTIC  GEOMETRY        [XI,  §  233 

233.  Polar  Equation.  Taking  the  focus  2<\  as  pole,  the 
perpendicular  from  Fi  toward  the  directrix  d^  as  polar  axis, 
and  putting  the  given  distance  F^D  =  q,  we  have  FiP  =  r, 
PQ  =  q  —  r  cos  <j>,  r  and  <^  being  the  polar  coordinates  of  any 
point  P  of  the  conic.  The  condition 
to  be  satisfied  by  the  point  P,  viz. 
FiP/PQ==e,  i.e.  F^P^e-PQ  becomes, 
therefore, 


e(g  — rcos  <^), 


whence      r  = 


1  4-  e  cos  <^  Fia.  96 


y 

L 

'% 

__« 

/f 

D 

X 

Fi 

c— -, 

k- 

iL 

? 

d, 

This  then  is  the  polar  equation  of  a  conic  if  the  focus  is  taken 
as  pole  and  the  perpendicular  from  the  focus  toward  the  directrix 
as  polar  axis. 

It  is  assumed  that  the  distance  q  between  the  fixed  point 
and  fixed  line  is  not  zero;  the  ratio  e,  i.e.  the  eccentricity  of 
the  conic,  may  be  any  positive  number. 

234.  Plotting  the  Conic.  By  means  of  this  polar  equation 
the  conic  can  be  plotted  by  points  when  e  and  q  are  given. 
Thus,  for  <^  =  0  and  <^  =  tt,  we  find  eq/{l  -\-  e)  and  eq/{l  —  e)  as 
the  intercepts  F^A^  and  F1A2  on  the  polar  axis ;  A^,  A2  are  the 
vertices.  For  any  negative  value  of  cf>  (between  0  and  —  tt) 
the  radius  vector  has  the  same  length  as  for  the  same  positive 
value  of  <fi.  The  segment  LL'  cut  off  by  the  conic  on  the  per- 
pendicular to  the  polar  axis  drawn  through  the  pole  is  called 
the  latus  rectum;  its  length  is  2  eg.  Notice  that  in  the  ellipse 
and  hyperbola,  i.e.  when  e  ^1,  the  vertex  Ai  does  not  bisect 
the  distance  FiD  (as  it  does  in  the  parabola),  but  that 

F^Ai/A^D  =  e. 


XI,  §  236] 


CONIC  SECTIONS 


225 


If  in  Fig.  96,  other  things  being  equal,  the  sense  of  the 
polar  axis  be  reversed,  we  obtain 
Fig.  97.  We  have  again  F^P=  r ;  but 
the  distance  of  P  from  the  directrix 
di  is  QP  =  q -\- r  cos  <f),  so  that  the 
polar  equation  of  the  conic  is  now : 

._ ^1 


1  —  e  cos  <f> 


y 

P. 

Q 

~L^ 

J) 

1  \ 

aA     \ 

di 

^ gr-U-      > 

^ 

Fig.  97 


235.  Classification  of  Conies.  For  e  =  1,  the  equations  of 
§§  233-234  reduce  to  the  equations  of  the  parabola  given  in 
§§  172,  173.  It  remains  to  show  that  for  e  <  1  and  e  >  1 
these  equations  represent  respectively  an  ellipse  and  a  hyper- 
bola as  defined  in  §§  204,  207. 

To  show  this  we  need  only  introduce  cartesian  coordi- 
nates and  then  transform  to  the  center^  i.e.  to  the  midpoint 
0  between  the  intersections  ^i,  A^  of  the  curve  with  the  polar 
axis. 

236.  Transformation  to  Cartesian  Coordinates.  The  equa- 
tion of  §  233, 

T  —  e{q  —  r  cos  <^) 

becomes  in  cartesian  coordinates,  with  the  pole  F^  as  origin 
and  the  polar  axis  as  axis  Ox  (Fig.  96) : 

VaJ^  -\-y^=  e{q  —  a;), 
or  rationalized : 

(1  -  e2).'c2  +  2  e  V  +  /  =  e'^'. 

The  midpoint  O  between  the  vertices  A-^,  A^  at  which  the 
curve  meets  the  axis  Ox  has,  by  §  234,  the  abscissa 

this  also  follows  from  the  cartesian  equation,  with  2/  =  0. 


226  PLANE  ANALYTIC  GEOMETRY        [XI,  §  237 

237.  Change  of  Origin  to  Center.  To  transform  to  paral- 
lel axes  through  this  point  0  we  have  to  replace  x  by 
X  —  e^q/(l  —  e^)  ;  the  equation  in  the  new  coordinates  is  there- 
fore 

and  this  reduces  to 


r' 


i.e. 


g2g2  ^2^2 

(1  -  6^)2       l-e2 

If  e  <  1  this  is  an  ellipse  with  semi-axes 

1  -  e^         vnr^' 

if  e  >  1  it  is  a  hyperbola  with  semi-axes 

238.  Focus  and  Directrix.  The  distance  c  (in  absolute  value) 
from  the  center  O  to  the  focus  F^  is,  as  shown  above,  for  the 
ellipse  „ 

c  =  — ^-  =  ae, 

1  —  e^ 

for  the  hyperbola 

e  —  \ 


The  distance  (in  absolute  value)  of   the  directrix  from  the 
center  0  is  for  the  ellipse,  since  g  =  a(l  —  e^)/e  =  a/e  —  ae  : 

and  for  the  hyperbola,  since  q  =  ae  —  a/e : 

OD  =  c-q  =  ae-ae-\--  =  -- 

e      e 


XI,  §238]  CONIC  SECTIONS  227 

It  is  clear  from  the  symmetry  of  the  ellipse  and  hyperbola 
that  each  of  these  curves  has  two  foci,  one  on  each  side  of  the 
center  at  the  distance  ae  from  the  center,  and  two  directrices 
whose  equations  are  a;  =  ±  a/e. 

EXERCISES 

1.   Sketch  the  following  conies  : 


2  +  3  COS  0  2  +  cos  0  1  —  2  cos  0 

2.  Sketch  the  following  conies  and  find  their  foci  and  directrices  : 

(a)  ic2  +  4  1/2  =  4,  (ft)  4  x2  +  1/2  _  4^ 

(c)  a:2  _  4  ^2  :^  4^  (ri)  4x2  -  2/2  =  4, 

(e)  16  x2  +  25  2/2  =  400,  (/)  9  a;2  -  16  2/2  =  144, 

(^)  9  a;2  -  16  y^  +  144  =  0,  (/t)  x2  - 1/2  =  2. 

3.  Show  that  the  following  equations  represent  ellipses  or  hyperbolas 
and  find  their  centers,  foci,  and  directrices : 

(a)  x2  +  32/2-2x+6?/  +  l  =0,         (6)  12x2  -  41/2  -  12x  -  9  =  0, 
(c)  5x2 +  y2  +  20x  + 15  =  0,  {d)  5x2-42/2  +  8?/  + 16  =  0. 

4.  Find  the  length  of  the  latus  rectum  of  an  ellipse  and  a  hyperbola 
in  terms  of  the  semi-axes. 

5.  Show  that  the  intersections  of  the  tangents  at  the  vertices  with 
the  asymptotes  of  a  hyperbola  lie  on  the  circle  about  the  center  passing 
through  the  foci. 

6.  Show  that  when  tangents  to  an  ellipse  or  hyperbola  are  drawn 
from  any  point  of  a  directrix  the  line  joining  the  points  of  contact  passes 
through  a  focus. 

7.  From  the  definition  (§  232)  of  an  ellipse  and  hyperbola,  show  that 
the  sum  and  difference  respectively  of  the  focal  radii  of  any  point  of  the 
conic  is  constant. 

8.  Find  the  locus  of  the  midpoints  of  chords  drawn  from  one  end  of  : 
(a)  the  major  axis  of  an  ellipse  ;  (&)  the  minor  axis. 

9.  The  eccentricity  of  an  ellipse  with  one  focus  and  corresponding 
directrix  fixed  is  allowed  to  vary;  show  that  the  locus  of  the  ends  of  the 
minor  axis  is  a  parabola. 

10.   Find  the  locus  of  §  232  when  the  fixed  point  lies  on  the  fixed  line. 


228 


PLANE  ANALYTIC  GEOMETRY        [XI,  §  239 


239.  The  Conies  as  Sections  of  a  Cone.  As  indicated  by 
their  name  the  conic  sections,  i.e.  the  parabola,  ellipse,  and 
hyperbola,  can  be  defined  as  the  curves  in  which  a  right  circu- 
lar cone  is  cut  by  a  plane  (§  231). 

In  Figs.  98,  99, 100,  Fis  the  vertex  of  the  cone,  ^  CVC'  =  2  a 
the  angle  at  its  vertex ;  OQ  indicates  the  cutting  plane,  CVC 
that  plane  through  the  axis  of  the 
cone  which  is  perpendicular  to  the 
cutting  plane.  The  intersection 
OQ  of  these  two  planes  is  evidently 
an  axis  of  symmetry  for  the  conic. 

The  conic  is  a  parabola,  ellipse, 
or  hyperbola,  according  as  OQ  is 
parallel  to  the  generator  VC  of  the 
cone  (Fig.  98),  meets  VC  at  a  point 
O'  belonging  to  the  same  half-cone 
as  does  O  (Fig.  99),  or  meets  FO' 
at  a  point  0'  of  the  other  half-cone  (Fig.  100). 
COQ  be  called  p,  the  conic  is 


Fig.  98 

If  the  angle 


a  parabola  if  /3  =  2  a  (Fig.  98), 
an  ellipse  if  ^  >  2  a  (Fig.  99), 
a  hyperbola  if  ^  <  2  a  (Fig.  100). 

In  each  of  the  three  figures  CO  represents  the  diameter  2  r 
of  any  cross-section  of  the  cone  {i.e.  of  any  section  at  right 
angles  to  its  axis).  We  take  O  as  origin,  OQ  as  axis  Ox,  so 
that  (Fig.  98)  OQ.  =  x,  QP=y  are  the  coordinates  of  any  point 
P  of  the  conic. 

As  QP  is  the  ordinate  of  the  circular  cross-section  CPC'P' 
we  have  in  each  of  the  three  cases  : 


y2^Qp2^CQ'QC\ 


XI,  §  241] 


CONIC  SECTIONS 


229 


240.   Parabola.     In  the  first  case  (Fig.  98),  when  y8  =  2  a  so 
that  OQ  is  parallel  to  VC,  the  expression 

X      OQ      OQ     ^ 

is  constant,  i.e.  the  same  at  whatever  distance  from  the  vertex 
we  may  take  the  cross-section  CPC'P'.  For,  QO  is  equal  to 
the  diameter   OB  =  ^r^  of  the  cross-section  through   0,  and 

CQ/OQ  =  CC'I  VC  =  2  r/r  esc  «  =  2  sin  a. 

Hence,  denoting  the  constant  r^  sin  a  by  p  we  have 

CQ 


OQ 


QC  =  4  ?o  sin  a  =4p. 


The  equation  of  the  conic  in  this  case,  referred  to  its  axis  OQ 
and  vertex  0,  is  therefore 

y^  =  4:px. 

Notice  that  as  p  =  Tq  sin  a  the  focus  is 
found  as  the  foot  of  the  perpendicular 
from  the  midpoint  of  OB  on  OQ. 

241.    Ellipse.     In   the    second    case 
(Fig.  99),  i.e.  when  ^  >  2  a,  if  we  put 

Oa  =  2a, 

it  can  be  shown  that 

f        ^     QP' 
x{2a-x)      OQ-QO' 


Fig.  99 


is  constant.     For  we  have  QP^  =  CQ  •  QC  and  from  the  tri- 
angles CQO,  QCa,  observing  that  ^  QaC  =  fi-2a: 

9Sl  =       si")g  QC'  ^  sin(^-2ct) 

OQ     sin(i7r-a)'  QO'     sin(^7r  +  a)' 


230 

whence 


PLANE  ANALYTIC  GEOMETRY        [XI,  §  241 


QP'      _  sin  )8sin(/3-2a) 


OQ  •  qa 


cos^  a 


an  expression  independent  of  the  position  of  the  cross-section 
CO. 
Denoting  this  positive  constant  by  h^,  we  find  the  equation 

y^  =  k'^x(2  a  —  x), 
(x-ay  ^     y'    ^^ 


i.e. 


ikaf 


which  represents  an  ellipse,  with  semi-axes  a,  ka  and  center 
(a,  0). 


242.  Hyperbola.  In  the  third  case 
(Fig.  100),  proceeding  as  in  the  second 
and  merely  observing  that  now 


qO'  =  -{2a  +  x\ 
we  find  the  equation 

y'^  =  k^x{2a-\-x), 


I.e. 


(x-ha)' 


(fca) 


which   represents   a   hyperbola,   with 
semi-axes  a,  ka  and  center  (—a,  0). 


Fig.  100 


243.  Limiting  Cases.  As  the  conic  is  an  ellipse,  hyperbola, 
or  parabola  according  as  /8  >  2  a,  <  2  a,  or  =  2  a,  it  appears 
that  ih.Q  parabola  can  be  regarded  as  the  limiting  case  of  either 
an  ellipse  or  a  hyperbola  whose  center  (the  midpoint  of  OCy) 
is  removed  to  infinity. 

On  the  other  hand,  if  in  the  second  case,  /?  >  2  a  (Fig.  99), 


XI.  §2431  CONIC  SECTIONS  231 

we  let  p  approach  tt,  or  if  in  the  third  case,  p  <2  a  (Fig.  100), 
we  let  p  approach  0,  the  cutting  plane  becomes  in  the  limit  a 
tangent  plane  to  the  cone.  •  It  then  has  in  common  with  the 
cone  the  points  of  the  generator  VC,  and  .these  only.  A  single 
straight  line  can  thus  appear  as  a  limiting  case  of  an  ellipse  or 
hyperbola. 

Finally  we  obtain  another  class  of  limiting  cases,  or  cases  of 
degeneration,  of  the  conies  if,  in  any  one  of  the  three  cases, 
we  let  the  cutting  plane  pass  through  the  vertex  V  of  the 
cone.  In  the  first  case,  (3  =  2  a,  the  cutting  plane  is  then  tan- 
gent to  the  cone  so  that  the  parabola  also  may  degenerate  into 
a  single  straight  line.  In  the  second  case,  ^  >  2  a,  if  /8  ^  tt, 
the  ellipse  degenerates  into  a  single  point,  the  vertex  V  of  the 
cone.  In  the  third  case,  /3  <  2  a,  if  /?  ^  0,  the  hyperbola  de- 
generates into  two  intersecting  lines. 

The  term  conic  section,  or  coiiic,  is  often  used  as  including 
these  limiting  cases. 

EXERCISES 

1.  For  what  value  of  /S  in  the  preceding  discussion  does  the  conic  be- 
come a  circle  ?  . 

2.  A  right  circular  cylinder  can  be  regarded  as  the  limiting  case  of  a 
right  circular  cone  whose  vertex  is  removed  to  infinity  along  its  axis 
while  a  certain  cross-section  remains  fixed.  The  section  of  such  a  cylin- 
der by  a  plane  is  in  general  an  ellipse ;  in  what  case  does  it  degenerate 
into  two  parallel  lines  ? 

3.  The  conic  sections  were  originally  defined  (by  the  older  Greek 
mathematicians,  in  the  time  of  Plato,  about  400  b.c.)  as  sections  of  a 
cone  by  a  plane  at  right  angles  to  a  generator  of  the  cone  ;  show  that  the 
section  is  a  parabola,  ellipse,  or  hyperbola  according  as  the  angle  2  a  at 
the  vertex  of  the  cone  is  =  |  tt,  <  |  tt,  >  |  tt. 

4.  Show  that  the  spheres  inscribed  in  a  right  circular  cone  so  as  to 
touch  the  cutting  plane  (Figs.  98,  99,  100)  touch  this  plane  at  the  foci  of 
the  conic. 


232  PLANE  ANALYTIC  GEOMETRY        [XI,  §  244 

PART   II.     REDUCTION   OF  GENERAL   EQUATION 

244.  Equations  of  Conies.  We  have  seen  in  the  two  pre- 
ceding chapters  that  hy  selecting  the  coordinate  system  in  a  con- 
venient way  the  equation  of  a  parabola  can  be  obtained  in  the 

simple  form  . 

y^=z4:px, 

that  of  an  ellipse  in  the  form 

a-'^b^-^' 
and  that  of  a  hyperbola  in  the  form 

a^      ¥ 

When  the  coordinate  system  is  taken  arbitrarily,  the  carte- 
sian equations  of  these  curves  will  in  general  not  have  this 
simple  form ;  but  they  will  always  be  of  the  second  degree. 
To  show  this  let  us  take  the  common  definition  of  these  curves 
(§  232)  as  the  locus  of  a  point  whose  distances  from  a  fixed 
point  and  a  fixed  line  are  in  a  constant  ratio.  With  respect  to 
any  rectangular  axes,  let  x^ ,  2/1  be  the  coordinates  of  the  fixed 
point,  ax  -{-by  -\-  c  =  0  the  equation  of  the  fixed  line,  and  e  the 
given  ratio.     Then  by  §§9  and  56  the  equation  of  the  locus  is 

or,  rationalized : 

{x  -  x,y  +  {y  -  y,y  =  -^  (ax  -hby-h  c)\ 
a^  -j-  0^ 

It  is  readily  seen  that  this  equation  is  always  of  the  second 
degree;  i.e.  that  the  coefiicients  of  a;^,  y"^,  and  xy  cannot  all 
three  vanish. 


XI,  §  246]       EQUATION  OF  SECOND  DEGREE  233 

245.  Equation  of  Second  Degree.  Conversely,  every  eq\ia- 
tion  of  the  second  degree,  i.e.  every  equation  of  the  form  (§  79) 
(1)  Ax"  +  2 Hxy -^ By""  +  2  Gx-\-2 Fy  ■\-  C  =  0, 
where  A,  H,  B  are  not  all  three  zero,  in  general  represents  a 
conic.  More  precisely,  the  equation  (1)  may  represent  an 
ellipse,  a  hyperbola,  or  a  parabola;  it  may  represent  two 
straight  lines,  different  or  coincident ;  it  may  be  satisfied  by 
the  coordinates  of  only  a  single  point;  and  it  may  not  be 
satisfied  by  any  real  point. 

Thus  each  of  the  equations 

a^  -  3  /  =  0,  xy  =  0 
evidently  represent  two  real  different  lines  ;  the  equation 

ic2_2a;  +  l  =  0 
represents  a  single  line,  or  as  it  is  customary  to  say,  two  coin- 
cident lines ;  the  equation 

a;2  +  ?/'  =  0 
represents  a  single  point,  while 

is  satisfied  by  no  real  point  and  is  sometimes  said  to  represent 
an  "imaginary  ellipse." 

The  term  conic  is  often  used  in  a  broader  sense  (compare  §  243) 
so  as  to  include  all  these  cases ;  it  is  then  equivalent  to  the 
expression  "locus  of  an  equation  of  the  second  degree.'^ 

It  will  be  shown  in  the  present  chapter  how  to  determine 
the  locus  of  any  equation  of  the  form  (1)  with  real  coefficients. 
The  method  consists  in  selecting  the  axes  of  coordinates  so  as 
to  reduce  the  given  equation  to  its  most  simple  form. 

246.  Translation  of  Axes.  The  transformation  of  the 
equation  (1)  to  its  most  simple  form  is  very  easy  in  the  par- 
ticular case  ichen  (1)  contains  no  term  in  xy,  i.e.  when  H  =  0. 
Indeed  it  suffices  in  this  case  to  complete  the  squares  in  x  and  y 
and  transform  to  parallel  axes. 


234  PLANE  ANALYTIC  GEOMETRY        [XI,  §  246 

Two  cases  may  be  distinguished: 

(a)  11=0,  A  =^  Oj  B  =^0,  so  that  the  equation  has  the  form 

(2)  Ax""  -\-  By^  -}-  2  Gx  +  2  Fy  +  C=  0. 

Completing  the  squares  in  x  and  y  (§  80),  we  obtain  an  equation 

of  the  form 

A  {X  -  hf  -\-B{y-  kf  =  K, 

where  ^  is  a  constant ;  upon  taking  parallel  axes  through  the 
point  {h,  k)  it  is  seen  thatxthe  locus  is  an  ellipse,  or  a  hyper- 
bola, or  two  straight  lines,  or  a  point,  or  no  real  locus,  accord- 
ing to  the  values  of  A,  B,  K. 

(h)  H=0,  and  either  ^=  0  or  J.=0,  so  that  the  equation  is 

(3)  Ax''  +  2Gx  +  2Fy-{-  0=0,  or  By' -h2Gx -\-2Fy  +  G=0. 
Completing  the  square  in  x  or  y,  we  obtain 

(x-hy=p(y-k),  or  (y -kf  =  q{x-h)', 

with  (h,  k)  as  new  origin  we  have  a  parabola  referred  to  vertex 
and  axis,  or  two  parallel  lines,  real  and  different,  coincident,  or 
imaginary. 

It  follows  from  this  discussion  that  the  absence  of  the  term  in 
xy  indicates  that,  in  the  case  of  the  ellipse  or  hyperbola,  its  axes, 
in  the  case  of  the  parabola,  its  axis  and  tangent  at  the  vertex,  are 
parallel  to  the  axes  of  coordinates. 

EXERCISES 

1.  Reduce  the  following  equations  to  standard  forms  and  sketch  the 
loci :  * 

(a)  2  2/2  _  3  a;  +  8 1/  +  11  =  0,  (b)  x^  +  ^y^  -  6x  +  iy +  6  =  0, 

(c)  6 x2  +  3  ?/2  -  4  a:  +  2 y  +  1  =  0,  (d)  x^  -  9y^  -  6x  +  ISy  =  0, 

(e)  9  a;2  +  9  2/2  -  36  x+6  ?/+  10  =  0,  (/)  2  j:^  -  iy"^  +  4  x  +  4y  -  1  =  0, 

(9)  x2  +  i/2_2x  +  22/-H3  =  0,  (h)  3x2  -  6x  +  y  +  6  =  0, 

(0  x2  -  ?/2  _  4  X  -  2  ?/  +  3  =  0,  ( j)  2  x2  -  5  X  +  12  =  0, 

(A;)  2  x2  -  5  X  -}-  2  =  0,  (0  y-^  -  4  y  +  4  =  0. 


XI,  §247]       EQUATION  OF  SECOND  DEGREE 


235 


2.  Find  the  equation  of  each  of  the  following  conies,  determine  the 
axis  perpendicular  to  the  given  directrix,  the  vertices  on  this  axis  (by 
division-ratio),  the  lengths  of  the  semi-axes,  and  make  a  rough  sketch 
in  each  case  : 

(a)  with  x  —  2  =  0  as  directrix,  focus  at  (6,  3),  eccentricity  | ; 

(i!>)  with  3x-|-4y— 6  =  0as  directrix,  focus  at  (5,  4) ,  eccentricity  | ; 

(c)  with  X  —  ?/  —  2  =  0as  directrix,  focus  at  (4,  0),  eccentricity  |. 

3.  Find  the  axis,  vertex,  latus  rectum,  and  sketch  thfe  parabola  with 
focus  at  (2,  —  2)  and  2a:  —  3 y  —  5  =  0  as  directrix  (see  Ex.  2). 

4.  Prove  the  statement  at  the  end  of  §  244. 

5.  Find  the  equation  of  the  ellipse  of  major  axis  5  with  foci  at  (0,  0) 
and  (3,  1). 

247.  Rotation  of  Axes.  If  the  right  angle  xOy  formed  by 
the  axes  Ox,  Oy  be  turned  about  the  origin  0  through  an 
angle  d  so  as  to  take  the  new  position  x^Oy^  (Fig.  101),  the 


relation  between  the  old  coordinates  OQ  =  x,  QP  =  y  of  any 
point  P  and  the  new  coordinates  OQi^x^,  QiP=yi  of  the 
same  point  P  are  seen  from  the  figure  to  be 

<  x  =  Xi  cos  0  —  yi  sin  0, 
[  y  =  x^  sin  0  +  .Vi  cos  6. 

By  solving  for  x^ ,  y^ ,  or  again  from  Fig.  101,  we  find 

j  x'l  =  X cos  6 -\-y  sin  9, 
\y^  =  —  X  ^\n  6  +  y  cos  6. 
If  the  cartesian  equation  of  any  curve  referred  to  the  axes 


(4) 


(4') 


236 


PLANE  ANALYTIC  GEOMETRY        [XI,  §  247 


Ox,  Oy  is  given,  the  equation  of  the  same  curve  referred  to  the 
new  axes  Ox^ ,  Oyi  is  found  by  substituting  the  values  (4)  for 
X,  y  in  the  given  equation. 

248.   Translation  and  Rotation.     To  transform  from  any 
rectangular  axes  Ox,  Oy  (Fig.  102)  to  any  other  rectangular 


y 

1 

y,. 

k 
h 

\    \ 

1                               X 

0 

jt 

Fia.  102 

axes   OxX^ ,    O^y-^ ,  we   have   to   combine   the   translation    00^ 
(§  13)  with  the  rotation  through  an  angle  6  (§  247). 

This  can  be  done  by  first  transforming  from  Ox,  Oy  to  the 
parallel  axes  Oyx\  O^y'  by  means  of  the  translation  (§  13) 

x  =  x^  -\-h, 

y  =  y'-\-  ^, 
and  then  turning  the  right  angle   x'Oiy'  through  the  angle 
0  =  x'OiXi ,  which  is  done  by  the  transformation  (§  247) 

x'  =  Xi  cos  6  —  yi  sin  6, 
2/'  =  iCi  sin  0  +  yi  cos  6. 

Eliminating  x',  y',  we  find 

x  =  XiC0s6  —  2/i  sin  6  -{-h, 
y  =  Xi  sin  0-\-yi  cos  $  +  lc. 
The  same  result  would  have  been  obtained  by  performing 
first  the  rotation  and  then  the  translation. 

It  has  been  assumed  that  the  right  angles  xOy  and  x^Oy^  are 
superposable ;  if  this  were  not  the  case,  it  would  be  necessary 
to  invert  ultimately  one  of  the  axes. 


(5) 


XI,  §  248]  EQUATION  OF  SECOND  DEGREE  237 

EXERCISES 

1.  Find  the  coordinates  of  each  of  the  following  points  after  the  axes 
have  been  rotated  about  the  origin  through  the  indicated  angle : 

(a)    (3,  4),  ^T.  (&)    (0,  5),i7r. 

(c)    (-3,  2),  <?  =  tan-i|.  (d)    (4,-3),^^- 

2.  K  the  origin  is  moved  to  the  point  (2,  -r- 1)  and  the  axes  then 

rotated  through  30"^,  what  will  be  the  new  coordinates  of  the  following 

points? 

(a)    (0,0).  (6)    (2,3).  (c)    (6,-1). 

3.  Find  the  new  equation  of  the  parabola  y^  =  i  ax  after  the  axes  have 
been  rotated  through :     (a)    ^tt     ,     (b)    ^tt    ,     (c)    tt     . 

—  4.   Show  that  the  equation  x^  +  y'^  =  a^  is  not  changed  by  any  rotation 
of  the  axes  about  the  origin.     Why  is  this  true  ? 

5.  Find  the  center  of  the  circle  {x—  a)^  +  y^  —a'^  after  the  axes  have 
been  turned  about  the  origin  through  the  angle  Q.  What  is  the  new 
equation  ? 

-  6.  For  each  of  the  following  loci  rotate  the  axes  about  the  origin 
through  the  indicated  angle  and  find  the  new  equation : 

/(a)   x2-i/2  +  2=0,  Itt.  (6)    x^-y'^  =  a\\ir. 

I    (c)   2/ =  mx  +  6,  0  =  tan-i m.      (d)    12x^  -  7 xy  -  12y^  =  0,  d  =  t&n-^l- 

Oi         0 

7.  Through  what  angle  must  the  axes  be  turned  about  the  origin  so 
that  the  circle  x^-^-y^  —  Sx  +  iy  —  6  =  0  will  not  contain  a  linear  term 
in  x? 

8.  Suppose  the  right  angle  XiOyi  (Fig.  101)  turns  about  the  origin  at 
a  uniform  rate  making  one  complete  revolution  in  two  seconds.  The 
coordinates  of  a  point  with  respect  to  the  moving  axes  being  (2,  1),  what 
are  its  coordinates  with  respect  to  the  fixed  axes  xOy  at  the  end  of  : 
(a)   i  sec.  ?     (b)   f  sec.  ?    (c)   1  sec.  ?     (d)    1^  sec.  ? 

9.  In  Fig.  101,  draw  the  line  OP,  and  denote  Z  QOP  by  <f>.  Divide 
both  sides  of  each  of  the  equations  (4)  by  OP  and  show  that  they  are 
then  equivalent  to  the  trigonometric  formulas  for  cos  (^  +  0)  and 
sin  (d  +  <p). 


238  PLANE  ANALYTIC  GEOMETRY        [XI,  §  249 

249.  Removal  of  the  Term  in  xy.  The  general  equation 
of  the  second  degree  (1),  §  245,  when  the  axes  are  turned  about 
the  origin  through  an  angle  ^  (§  247),  becomes  : 

A  (a^i  cos  6  —  yi  sin  fff 

+  2  H(x^  cos  d~  2/i  sin  6)  {x^  sin  O  +  yi  cos  6) 
+  J5(a;i  sin  (9  +  2/1  cos  ^)2 
+  2  G{Xy^  cos  d  —  yi  sin  $) 
+  2 F{x^  sin  ^  +  2/i  cos  ^)  +  (7=  0. 
This  is  an  equation  of  the  second  degree  in  x^^  and  y^  in 
which  the  coefficient  of  x^y^  is  readily  seen  to  be 

—  2^cos^sind  +  ^JBsin^cos^  +  2^(cos2(9-sin2^) 

=  {B-  A)  sin  2  ^  +  2  fi^cos  2  6. 

It  follows  that  if  the  axes  be  turned  about  the  origin 
through  an  angle  6  such  that 

(JS -^)  sin  2  ^H- 2  ITcos  2  ^  =  0, 

i.e.  such  that  r 

2H 


(6)  tan  2^ 


A-B' 


the  equation  referred  to  the  new  axes  will  contain  no  term  in 
x^y^  and  can  therefore  be  treated  by  the  method  of  §  246. 
According  to  the  remark  at  the  end  of  §  246  this  means 
that  the  new  axes  Oa^i,  Oyi,  obtained  by  turning  the  original 
axes  Ox,  Oy  through  the  angle  0  found  from  (6),  are  parallel 
to  the  axes  of  the  conic  (or,  in  the  case  of  the  parabola,  to  the 
axis  and  the  tangent  at  the  vertex). 

The  equation  (6)  can  therefore  be  used  to  determine  the 
directions  of  the  axes  of  the  conic;  but  the  process  just  indicated 
is  generally  inconvenient  for  reducing  a  numerical  equation  of 
the  second  degree  to  its  most  simple  form  since  the  values  of 
cos  0  and  sin  6  required  by  (4)  to  obtain  the  new  equation  are 
in  general  irrational. 


XI,  §250]      EQUATION  OF  SECOND  DEGREE  239 

EXERCISES 

1.  Through  what  angle  must  the  axes  be  turned  about  the  origin  to 
remove  the  term  in  xy  from  each  of  the  following  equations  ? 

(a)  3;:c2+2\/3a;?/+?/2_3a;+4?/-10=0.      (6)  x;^  +  2y/Ixy  +  1  y'^-\^  =  0. 
(c)   2x2- 3a;?/ +  2^/2 +  x- 2/ +7=0.         {d)xy  =  2a'^. 

2.  Reduce  each  of  the  following  equations  to  one  of  the  forms  in  §  244  : 
(a)  xy  =  -%  (6)    6  x2  -  5  xy  -  6  2/2  =  0. 

(c)   3x2-10x^  +  32/2  +  8  =  0.  {d)    13x2  -  lOxy  +  13^2  _  72  =  o. 

250.  Transformation  to  Parallel  Axes.  To  transform  the 
general  equation  of  the  second  degree  (1),  §  245,  to  parallel 
axes  through  any  point  (x^,  y^),  we  have  to  substitute  (§  13) 

x  =  x'  +  Xq,     y=y'  +  yo, 

the  resulting  equation  is 

Ax''  +  2  Hxy  4-  By''  +  2  (Ax,  +  Hy,  +  G)  a/ 

+  2(J7a^o  +  52/o  +  i^)/  +  C"  =  0, 

where  the  new  constant  term  is 

(7)  e  =  Ax,'  +  2Hx,y,^By,'  +  2Gx,-^2Fy,-\-a 

It  thus  appears  that  after  any  trarislation  of  the  coordinate 
system : 

(a)  the  coefficients  of  the  terms  of  the  second  degree  remain 
unchanged ; 

(b)  the  new  coefficients  of  the  terms  of  the  first  degree  are 
linear  functions  of  the  coordinates  of  the  new  origin ; 

(c)  the  new  constant  term  is  the  result  of  substituting  the 
coordinates  of  the  new  origin  in  the  left-hand  member  of  the 
original  equation. 


240  PLANE  ANALYTIC  GEOMETRY        [XI,  §  251 

251.  Transformation  to  the  Center.  The  transformed  equa- 
tion will  contain  no  terms  of  the  first  degree,  i.e.  it  will  be  of 
the  form  _,  - 

(8)  Ax"'  +  2  H^y'  +  By"'  -^  C  =  0, 

if  we  can>gelect  the  new  origin  {x^^  y^)  so  that 

.gs       /  Ax,-{-Hy,+  G  =  0, 

^^  Hx,  +  By,  +  F^O. 

This  is  certainly  possible  whenever 

A     H 


and  we  then  find  : 

no^  X  -FH-GB        ^OH-FA 

^     ^  "^      AB-H^'  -^^      AB- 11^ 


As  the  equation  (8)  remains  unchanged  when  x',  y'  are 
replaced  by  —  x\  —  y',  respectively,  the  new  origin  so  found  is 
the  center  of  the  curve  (§  224).  The  locus  is  therefore  in 
this  case  a  central  conic,  i.e.  an  ellipse  or  a  hyperbola;  but  it 
may  reduce  to  two  straight  lines  or  to  a  point  (see  §  254).  It 
might  be  entirely  imaginary,  viz.  if  ^=  0 ;  but  the  case  when 
11=0  has  already  been  discussed  in  §  246. 

We  shall  discuss  in  §  256  the  case  in  which  AB  —  H^  =  0. 

252.  The  Constant  Term  and  the  Discriminant.  The  cal- 
culation of  the  constant  term  C  can  be  somewhat  simplified 
by  observing  that  its  expression  (7)  can  be  written 

C  =(Ax,-{-  H7j,  +  G)x,-h(Hx,  +  By,  +  F)y,-\-  Gx,-\-  Fy,-^  C, 
i.e.,  owing  to  (9), 

(11)  C'=:Gx,-\-Fy,-^a 

If  we  here  substitute  for  x^,  y^  their  values  (10)  we  find  : 
GFII  -  G^B  +  FGH  -  F^A  -f-  ABC  -  H'C 


C 


AB-H' 


XI,  §  253]      EQUATION  OF  SECOND  DEGREE  241 

The  numerator,  which  is  called  the  discriminant  of  the  equa- 
tion of  the  second  degree  and  is  denoted  by  D,  can  be  written 
in  the  form  of  a  symmetric  determinant,  viz. 

A     H    G 
D=  H    B     F  ' 
G     F     C 

If  we  denote  the  cofactors  of  this  determinant  by  the  corre- 
sponding small  letters,  we  have 

^0  —      J    ?/o  —        '     ^    — 

C  C  C 

Notice  that  the  coefficients  of  the  equations  (9),  which  deter- 
mine the  center,  are  given  by  the  first  two  rows  of  Z>,  while  the 
third  row  gives  the  coefficients  of  C"  in  (11). 

253.  Homogeneous  Function  of  Second  Degree.  The  nota- 
tion for  the  coefficients  in  the  equation  of  the  second  degree  arises  from 
the  fact  that  the  left-hand  member  of  this  equation  can  be  regarded  as 
the  value  for  2;  =  1  of  the  general  homogeneous  function  of  the  second 
degree,  viz. 

/(a;,  y,  z)  =  Ax'^  +  By'^  -\-  Cz^  +  2  Fyz  +  2Gzx-{-2  Hxy. 

If  in  this  function  x  alone  be  regarded  as  variable  while  y  and  z  are 
treated  as  constants,  the  derivative  with  respect  to  x  is 

fj  =2{Ax-\-Hy  +  Gz)', 

if  y  alone,  or  z  alone,  be  regarded  as  variable,  we  find  similarly 

fy'  =  2{Hx  +  By  +  Fz), 
f^'  =  2{Gx  +  Fy+Cz). 

These  partial  derivatives  of  the  homogeneous  function  /(x,  y,  z)  with 
respect  to  a;,  ?/,  2,  respectively,  are  linear  homogeneous  functions  of  aj,  y,  z^ 
and  it  is  at  once  verified  that 

i.e.  the  homogeneous  function  of  the  second  degree  is  equal  to  half  the  sum 
of  the  products  of  its  partial  derivatives  by  x,  y,  z. 


A    H 

G 

H    B 

F 

G     F 

C 

242  PLANE  ANALYTIC  GEOMETRY        [XI,  §  253 

The  left-hand  members  of  the  equations  (0)  are  IfJixQ,  yo ,  1), 
i/i/'C^o,  Vqi  !)•  Hence  the  equations  for  the  center  can  he  obtained  by 
differentiating  /(x,  y,  0),  or  what  amounts  to  the  same,  the  left-hand 
member  of  the  equation  of  the  second  degree,  with  respect  to  x  alone  and 
y  alone. 

The  symmetric  determinant 

D 

formed  of  the  coefificients  of  ^/x',  \fy  -,  \fz  is  called  the  discriminant  of 
f(x,  y,  z) ;  and  this  is  also  the  discriminant  of  the  equation  of  the  second 
degree  (§252).  As  f=  i(fjx  + fy'y -\-f,'z)  andfJ{xo,  yo,  1)  =  0, 
fy'(oco  ,  yo,  1)  =  0  it  follows  that 

C  =f(xo  ,  1/0  ,  1)  =  lf^'(xo  ,  yo  ,  1)  =  Gxo  -{-Fyo  +  C. 

^  254.  Straight  Lines.  After  transforming  to  the  center,  i.e. 
obtaining  the  equation  (8),  we  must  distinguish  two  cases 
according  as  G'  =  0  or  C'=^0.  The  condition  C'  =  0  means 
by  (7)  that  the  center  lies  on  the  locus ;  and  indeed  the  homo- 
geneous equation 

represents  two  straight  lines  through  the  new  origin  (a^o ,  2/0) 
(§  59).  The  separate  equations  of  these  lines,  referred  to 
the  new  axes,  are  found  by  factoring  the  left-hand  member. 
As  we  here  assume  (§  251)  that  AB  —  H^=^0,  and  H^O,  the 
lines  can  only  be  either  real  and  different,  or  imaginary.  In 
the  latter  case  the  point  (a;„ ,  y^)  is  the  only  real  point  whose 
coordinates  satisfy  the  original  equation. 

255.   Ellipse  and  Hyperbola.     If  C  =^0  we  can  divide  (8) 
by  —  C  so  that  the  equation  reduces  to  the  form 
(12)  ax'  +  2hxy-\-by-  =  l. 

This  equation  represents  an  ellipse  or  a  hyperbola  (since  we 
assume  h=^0).  The  axes  of  the  ellipse  or  hyperbola  can  be 
found  in  magnitude  and  direction  as  follows. 


XI,  §255]      EQUATION  OF  SECOND  DEGREE 


243 


Fig.  103 


If  an  ellipse  or  hyperbola,  with  its  center,  be  given  graphi- 
cally, the  axes  can  be  constructed  by  inter- 
secting the  curve  with  a  concentric  circle 
and  drawing  the  lines  from  the  center  to 
the  intersections;  the  bisectors  of  the 
angles  between  these  lines  are  evidently 
the  axes  of  the  curve  (Fig.  103). 

The  intersections  of  the  curve  (12)  with 
a  concentric  circle  of  radius  r  are  given  by 
the  simultaneous  equations 

aa;2  H-  2  lixy  -f  h]f-  =  1,  ^-{-if^r^'^ 

dividing  the  second  equation  by  r^  and  subtracting  it  from  the 
first,  we  have 

(13)  ^a-iy +  2/10^7/ -f  ^6 -iy^  =  0. 

This  homogeneous  equation  represents  two  straight  lines 
through  the  origin,  and  as  the  equation  is  satisfied  by  the 
coordinates  of  the  points  that  satisfy  both  the  preceding  equa- 
tions, these  lines  must  be  the  lines  from  the  origin  to  the  inter- 
sections of  the  circle  with  the  curve  (12).     If  we  now  select  r 


(14) 


(14-) 


a  — 


so  as  to  make  thejbwo  lines  (13)  coincide,  they  will  evidently 
coincide  with  one  or  the  other  of  the  axes  of  the  curve  (12). 
The  condition  for  equal  roots  of  the  quadratic  (13)  in  y/x  is 

This  equation,  which  is  quadratic  in  l/r^  and  can  be  written 

determines  the  lengths  of  the  axes.  If  the  two  values  found  for 
ir  are  both  positive,  the  curve  is  an  ellipse ;  if  one  is  positive 


'-(aH-6)i  +  a6-7i2  =  0, 


244  PLANE  ANALYTIC  GEOMETRY        [XI,  §  255 

and  the  other  negative,  it  is  a  hyperbola ;  if  both  are  negative, 
there  is  no  real  locus. 

Each  of  the  two  values  of  1/r^  found  from  (14'),  if  substi- 
tuted in  (13),  makes  the  left-hand  member,  owing  to  (14),  a 
complete  square.     Tlie  equations  of  the  axes  are  therefore 


\a-h^±yjf>-^y  =  0, 


or,  multiplying  by  Va  —  l/r^  and  observing  (14)  : 
a ]x  -h  hy  =  0. 

256.  Parabola.  It  remains  to  discuss  the  case  (§  251)  of  the 
general  equation  of  the  second  degree. 

Ax''  +  2  Hxy  +  By^  +  2Gx  +  2  Fy  +0  =  0, 
in  which  we  have  ^^  _  jj2  _  q 

This  condition  means  that  the  terms  of  the  second  degree  form 
a  perfect  square  : 

Ax""  +  2  Hxy  +  By^  =  (VAx  +  VSyy. 
Putting  V^  =  a  and  V^  =  6  we  can  write  the  equation  of  the 
second  degree  in  this  case  in  the  form 

(1 5)  (ax  +  byf  =  -2Gx-2Fy-a 

If  G  and  F  are  both  zero,  this  equation  represents  two  parallel 
straight  lines,  real  and  different,  real  and  coincident,  or  im- 
aginary according  as  (7  <  0,  C  =  0,  (7  >  0. 

If  G  and  F  are  not  both  zero,  the  equation  (15)  can  be  inter- 
preted as  meaning  that  the  square  of  the  distance  of  the  point 
(x,  y)  from  the  line 

(16)  ax-}-by  =  0 

is  proportional  to  the  distance  of  (cc,  y)  from  the  line 

(17)  2Gx  +  2Fy-^C=0. 

Hence  if  these  lines  (16),  (17)  happen  to  be  at  right  angles,  the 


XI,  §  256]      EQUATION  OF  SECOND  DEGREE  245 

locus  of  (15)  is  Si  parabola,  having  the  line  (16)  as  axis  and  the 
line  (17)  as  tangent  at  the  vertex. 

But  even  when  the  lines  (16)  and  (17)  are  not  at  right  angles 
the  equation  (15)  can  be  shown  to  represent  a  parabola.  For 
if  we  add  a  constant  k  within  the  parenthesis  and  compensate 
the  right-hand  member  by  adding  the  terms  2  aJcx  -f-  2  bky  +  7c^, 
the  locus  of  (15)  is  not  changed ;  and  in  the  resulting  equation 

(18)  (ax  +  by  +  kf  =  2(ak  -  G)x  -f  2(bk  -  F)y  -{-k^-C 
we  can  determine  k  so  as  to  make  the  two  lines 

(19)  ax  +  by-^k  =  0, 

(20)  2(ak  -  G)x  +  2{bk  -F)y  +  k^-C=0 

perpendicular.     The  condition  for  perpendicularity  is 

a{ak -  G) -\-b{bk - F)  =  0, 
whence 

(21)  k^^^±^. 

With  this  value  of  k,  then,  the  lines  (19),  (20)  are  at  right 
angles ;  and  if  (19)  is  taken  as  new  axis  Ox  and  (20)  as  new 
axis  Oy^  the  equation  (18)  reduces  to  the  simple  form 

y^  =  px. 
The  constant  p,  i.e.  the  latus  rectum  of  the  parabola,  is  found 
by  writing  (18)  in  the  form 
fax  4-  &y  +  ^\_ 


2 V(afc  -  Gf  +  i^k  -  Fy-  2{ak  -  G)x  +  2{bk~F)y-\-k''-  (7. 

«'+&'  *       2V(ak-Gy  +  (bk-Fy 

hence 


Substituting  for  k  its  value  (21)  we  can  reduce  it  to 

^  2(aF-bG) 
{a'-\-¥)^ 


246  PLANE  ANALYTIC  GEOMETRY        [XI,  §  256 

EXERCISES 

1.  Find  the  equation  of  each  of  the  following  loci  after  transforming 
to  parallel  axes  through  the  center : 

(a)  Sx^-4xy-y'^-Sx-iy  +  7  =  0. 
(6)  6  x^  +  6  xy  -\-  y^  +  6  X  -  4:  y  —  6  =  0. 

(c)  2  x^  -\-  xy  -  6  y^  —  7  X  —  7  y  -\-  5  =  0. 

(d)  x'^  -  2  xy  -  y^  -\-  i  X  -  2y  -  8  =  0. 

2.  Find  that  diameter  of  the  conic  Sx^  —  2xy—4:y'^+6x—4:y  -^-2=0 
(a)  which  passes  through  the  origin,  (&)  which  is  parallel  to  each  co- 
ordinate axis. 

3.  For  what  values  of  k  do  the  following  equations  represent  straight 
lines  ?    Find  their  intersections. 

ia)  2x^  -  xy-Sy'^-6x  +  19y  +  k  =  0. 
(6)  kx^  +  2  xy  -{-  y^  -  X  -  y  -  6  =  0. 

(c)  S  x:^  -  4  xy  +  ky^  +  S  y  -  S  =  0. 

(d)  x^-^2y^  +  6x-4y  +  k  =  0. 

4.  Show  that  the  equations  of  conjugate  hyperbolas  x^/a^—y^/b'^=  ±1 
and  their  asymptotes  x^/a^—y'^/b^  =  0,  even  after  a  translation  and  rota- 
tion of  the  axes,  will  differ  only  in  the  constant  terms  and  that  the  con- 
stant term  of  the  asymptotes  is  the  arithmetic  mean  between  the  constant 
terms  of  the  conjugate  hyperbolas. 

5.  Find  the  asymptotes  and  the  hyperbola  conjugate  to 

2x^  —  xy  -  15y-^  +  X+  19y  +  16=0. 

6.  Find  the  hyperbola  through  the  point  (—2,  1)  which  has  the  lines 
2x  —  y+l  =  0,  3x4-2?/  —  6  =  0  as  asymptotes.  Find  the  conjugate 
hyperbola. 

7.  Show  that  the  hyperbola  xy  =  a^  is  referred  to  its  asymptotes  as 
coordinate  axes.  Find  the  semi-axes  and  sketch  the  curve.  Find  and 
sketch  the  conjugate  hyperbola. 

8.  The  volume  of  a  gas  under  constant  temperature  varies  inversely 
as  the  pressure  (Boyle's  law),  i.e.  vp  =  c.  Sketch  the  curve  whose  ordi- 
nates  represent  the  pressure  as  a  function  of  the  volume  for  different 
values  of  c  ;  e.g.  take  c  =  1,  2,  3. 

9.  Sketch  the  hyperbola  (x  —  a)(y  —  b)  =  c^  and  its  asymptotes.  In- 
terpret the  constants  a,  b,  c  geometrically. 


XI,  §256]     EQUATION  OF  SECOND  DEGREE  247 

10.  Sketch  the  hyperbola  xy-\-Sy  —  6  =  0  and  its  asymptotes. 

11.  Find  the  center  and  semi-axes  of  the  following  conies,  write  their 
equations  in  the  most  simple  form,  and  sketch  the  curves  : 

^(a)  6  x^  -  6  xy  +  5  y^  +  12y/2  X  -  W2y  +  8  =  0. 
^ (6)  x2  -  6\/8 xi/  -  6  ^/2  -  16  =  0.     (c)  x'^ -{- xy  -{- y^  -  Sy  +  Q  =  0. 
(d)  13x^-QV3xy  +  7y^-M  =  0. 
^    (e)   2  x2  -  4  X2/  +  ?/2  4-  2  X  -  4  ?/  -  f  =  0. 
^  (/)  3  x2  +  2x2/ +  2/2  +  6x  +  4  ?/ +  I  =  0. 

12.  Sketch  the  following  parabolas  : 

(a)  x2  _  2V3xy  +  3  «/2  -  6  V3  x  -6y  =  0. 
(&)  x2  -  6  xy  +  9  «/2  -  3  X  +  4y  -  1  =  0. 

13.  Show  that  the  following  combinations  of  the  coefficients  of  the 
general  equation  of  the  second  degree  are  invariants  (i.e.  remain  un- 
changed) under  any  transformation  from  rectangular  to  rectangular  axes : 

(a)  A  +  B.  (6)  AB  -  H^.  (c)   (A  -  ^)2  +  4  iI2. 

14.  Show  that  x2  +  y^  =  a^  represents  a  parabola.     Sketch  the  locus. 

15.  Find  the  parabola  with  x  +  y  =  0  as  directrix  and  (^  a,  |  a)  as 
focus. 

16.  Let  five  points  A,  B,  C,  D,  E  be  taken  at  equal  intervals  on  a 
line.  Show  that  the  locus  of  a  point  P  such  that  AP  ■  EP  =  BP  •  DP  is 
an  equilateral  hyperbola.     (Take  G  as  origin.) 

17.  The  variable  triangle  AQB  is  isosceles  with  a  fixed  base  AB. 
Show  that  the  locus  of  the  intersection  of  the  line  AQ  with  the  perpen- 
dicular to  QB  through  B  is  an  equilateral  hyperbola. 

18.  Let  ^  be  a  fixed  point  and  let  Q  describe  a  fixed  line.  Find  the 
locus  of  the  intersection  of  a  line  through  Q  perpendicular  to  the  fixed 
line  and  a  line  through  A  perpendicular  to  AQ. 

19.  Find  the  locus  of  the  intersection  of  lines  drawn  from  the  extrem- 
ities of  a  fixed  diameter  of  a  circle  to  the  ends  of  the  perpendicular 
chords. 

20.  Show  by  (14'),  §255,  that  if  the  equation  of  the  second  degree 
represents  an  ellipse,  parabola,  hyperbola,  we  have,  respectively, 

^S  -  If  2  ^  0,  =  0,  <  0.    . 


CHAPTER  XII 

HIGHER  PLANE   CURVES 

PART   I.     ALGEBRAIC   CURVES 

257.  Cubics.  It  has  been  shown  (§  30)  that  every  equation 
of  the  first  degree, 

H-  a^x  +  6i2/  =  0, 

represents  a  straight  line;  and  (§  245)  that  every  equation  of 
the  second  degree, 

Oo 
+  a^x  +  biy 
+  a^"^  4-  h^y  +  C22/2  =  0, 
either  represents  a  conic  or  is  not  satisfied  by  any  real  points. 
The  locus  represented  by  an  equation  of  the  third  degree, 

4-  a^x^  +  h^y  -f-  c^"^ 
+  a^a?  -f  h^'^y  +  c^xy"^  +  d^^=  0, 

I.e.  the  aggregate  of  all  real  points  whose  coordinates  x,  y  satisfy 
this  equation,  is  called  a  cubic  curve. 

Similarly,  the  locus  of  all  points  that  satisfy  any  equation  of 
the  fourth  degree  is  called  a  quartic  curve;  and  the  terms  quintic, 
sextic,  etc.,  are  applied  to  curves  whose  equations  are  of  the 
Jifthj  sixth,  etc.,  degrees. 

Even  the  cubics  present  a  large  variety  of  shapes;  still 
more  so  is  this  true  of  higher  curves.  We  shall  not  discuss 
such  curves  in  detail,  but  we  shall  study  some  of  their  properties. 

248 


XII,  §258]  ALGEBRAIC  CURVES  249 

258.   Algebraic  Curves.     The  general  form  of  an  algebraic 
equation  of  the  .nth  degree  in  x  and  y  is 

+  a^x  -{-b^y 
(1)  +  a^^  -f-  b^xy  +  Cojy^ 

-\-a^-{-  b^x^y  4-  c^y"^ + d^ 


+  a^x""  +  b^x^-^y  +  ...4-  Kxtj^-'^+  l^y""  =  0. 

The  coefficients  are  supposed  to  be  any  real  numbers,  those  in 
the  last  line  being  not  all  zero.  The  number  of  terms  is  not 
more  than  1  +  2  +  3  +  ...  +(n  +  1)  =  i(n  +  l)(n  +  2). 

If  the  cartesian  equation  of  a  curve  can  be  reduced  to  this 
form  by  rationalizing  and  clearing  of  fractions,  the  curve  is 
called  an  algebraic  curve  of  degree  n. 

An  algebraic    curve   of  degree  n  can  be  intersected  by  a 

straight  line, 

Ax-\-  By-h  C=0, 

in  not  more  than  n  points.  For,  the  substitution  in  (1)  of  the 
value  of  y  (or  of  x)  derived  from  the  linear  equation  gives  an 
equation  in  x  (or  in  y)  of  a  degree  not  greater  than  n ;  this 
equation  can  therefore  have  not  more  than  n  roots,  and  these 
roots  are  the  abscissas  (or  ordinates)  of  the  points  of  intersec- 
tion. 

We  have  already  studied  the  curves  that  represent  the  poly- 
nomial function 

y=ao+ aiX-{-a^-\ hotna^"; 

such  a  curve  is  an  algebraic  curve,  but  it  is  readily  seen  by 
comparison  with  the  preceding  equation  that  this  equation  is 
of  a  very  special  type,  since  it  contains  no  term  of  higher  de- 
gree than  one  in  y.  Such  a  curve  is  often  called  a  parabolic 
curve  of  the  nth  degree. 


250 


PLANE  ANALYTIC  GEOMETRY      [XII,  §  259 


259.  Transformation  to  Polar  Coordinates.  The  cartesian 
equation  (1)  is  readily  transformed  to  polar  coordinates  by  sub- 
stituting 

X  =  r  cos  <^,     y  =  r  sin  <^ ; 

it  then  assumes  the  form : 

+  (aj  cos  4*  -\-hi  sin  <^)?' 
(2)  -f  (as  cos^  <\>-{-h.2  cos  <^  sin  <J!)  +  Cg  sin^  <l>)r^ 

+  (ttg  cos^  <^  H-  &3  cos^  <^  sin  <^  -h  Cg  cos  <j>  sin^  <^  +  c^s  sin'  <^)r^ 


+  (a„  cos"  <^ + &„  cos"~^  <^  sin  <j>  + 


+fc„cos<^sin"-^  <^-f/^  sin'*  </>)?•» 
=  0. 


If  any  particular  value  be  assigned  to  the  polar  angle  <^,  this 
becomes  an  equation  in  r  of  a 
degree  not  greater  than  n.  Its 
roots  ri,  r^,'"  represent  the  in- 
tercepts OPi,  OP2,  "  (Fig.  104) 
made  by  the  curve  (2)  on  the  line 
y  =  tan  <^  •  x.  Some  of  these 
roots  may  of  course  be  imaginary, 
and  there  may  be  equal  roots.  Fig.  104 

260.  Curve  through  the  Origin.  The  equation  in  r  has  at 
least  one  of  its  roots  equal  to  zero  if,  and  only  if,  the  constant 
term  ao  is  zero.  Thus,  the  necessary  and  sufficient  condition  that 
the  origin  0  he  a  point  of  the  curve  is  aQ  =  0. 

This  is  of  course  also  apparent  from  the  equation  (1)  which 
is  satisfied  by  ic  =  0,  2/  =  0  if,  and  only  if,  ao  =  0. 

261.  Tangent  Line  at  Origin.  The  equation  (2)  has  at 
least  two  of  its  roots  equal  to  zero  if  ao  =  0  and  ai  cos  <^  + 
61  sin  <f>  =  0.     If  ai  and  bi  are  not  both  zero,  the  latter  condition 


XII,  §  263] 


ALGEBRAIC  CURVES 


251 


can  be  satisfied  by  selecting  the  angle  <^  properly,  viz.  so  that 
tan<^  =  -^. 


The  line  through  the  origin  inclined  at  this  angle  <^  to  the 
polar  axis  is  the  tangent  to  the  curve  at  the  origin  0  (Fig.  105). 
Its  cartesian  equation  is  2/  =  tan  <^'X  —  —  (a^/h^x,  i.e. 

(3)  a^x  H-  h^y  =  0. 

Thus,  if  tto  =  0  while  Oi ,  by  are  not  both  zero,  the  curve  has 
at  the  origin  a  single  tangent ;  the  origin  0  is  therefore  called 
a  simple,  or  ordinary,  point  of  the  curve. 
In  other  words,  if  the  lowest  terms  in 
the  equation  (1)  of  an  algebraic  curve 
are  of  the  first  degree,  the  origin  is  a 
simple  point  of  the  curve,  and  the  equa- 
tion of  the  tangent  at  the  origin  is  ob- 
tained by  equating  to  zero  the  terms  of 
the  first  degree.  Fig.  105 

262.  Double  Point.  The  condition  aicos  <^  +  ^i  sin  <^  =  0 
necessary  for  two  zero  roots  is  also  satisfied  if  «!  =  0  and  &i  =  0 ; 
indeed,  it  is  then  satisfied  whatever  the  value  of  <^.  Hence,  if 
a^  =  0,  %  =  0,  61  =  0,  the  equation  (2)  has  at  least  two  zero 
roots  for  any  value  of  <^.  If  in  this  case  the  terms  of  the 
second  degree  in  (1)  do  not  all  vanish,  the  curve  is  said  to 
have  a  double  point  at  the  origin.  Thus,  the  origin  is  a  double 
l)oint  if,  and  only  if,  the  loivest  terms  in  the  equation  (1)  are  of 
the  second  degree. 

263.  Tangents  at  a  Double  Point.  The  equation  (2)  will 
have  at  least  three  of  its  roots  equal  to  zero  if  we  have  ao  =  0, 
ttj  =  0,  61  =  0  and 

Oa  cos'^  <^  4-  62  cos  <^  sin  <^  +  Cg  sin^  <^  =  0. 


252 


PLANE  ANALYTIC   GEOMETRY      [XII,  §  263 


K  a^,  63,  C2  are  not  all  zero,  we  can  find  two  angles  satisfying 
this  equation  which  may  be  real  and  different,  or  real  and 
equal,  or  imaginary.  The  lines  drawn  at  thjese  angles  (if  real) 
through  the  origin  are  the  tangents  at  the  double  point. 

Multiplying  the  last  equation  by  7^  and  reintroducing  carte- 
sian coordinates  we  obtain  for  these  tangents  the  equation 


(4) 


tta^J^  +  b^y  -\-  c^y^  =  0. 


Thus,  if  the  loivest  terms  in  the  equation  (1)  are  of  the  second 
degree^  the  origin  is  a  double  point,  and  these  terms  of  the  second 
degree  equated  to  zero  represent  the  tangents  at  the  origin. 


264.  Types  of  Double  Point,  (a)  If  the  two  lines  (4)  are 
real  and  different,  the  double  point  is 
called  a  node  or  crunode ;  the  curve  then 
has  two  branches  passing  through  the 
origin,  each  with  a  different  tangent 
(Fig.  106).  ^ 

(b)  If  the  lines  (4)  are  coincident,  i.e. 
if  ttg^  +  b<p:y  +  c^y"^  is  a  complete  square,  Fig.  106 

the  double  point  is  called  a  cusp,  or  spinode;  the  curve  then 
has   ordinarily  two  real  branches  tangent  to 
one  and  the  same  line  at  the  origin  (Fig.  107 
represents  the  most  simple  case). 

(c)  If  the  lines  (4)  are  imaginary,  the 
double  point  is  called  an  isolated  point,  or 
an  acnode;  in  this  case,  while  the  coordi- 
nates 0,  0  of  the  origin  satisfy  the  equation 
of  the  curve,  there  exists  about  the  origin 
a  region  containing  no  other  point  of  the 
curve,  so  that  no  tangents  can  be  drawn 
through  the  origin  (Fig.  108). 


J^ 


FiQ.  107 


Fig.  108 


XII,  §265]  ALGEBRAIC  CURVES  253 

It  should  be  observed  that,  for  curves  of  a  degree  above 
the  third,  the  origiu  in  case  (b)  may  be  an  isolated  point ;  this 
will  be  revealed  by  investigating  the  higher  terms  (viz.  those 
above  the  second  degree). 

265.  Multiple  Points.  It  is  readily  seen  how  the  reasoning 
of  the  last  articles  can  be  continued  although  the  investigation 
of  higher  multiple  points  would  require  further  discussion. 
The  result  is  this :  If  in  the  equation  of  an  algebraic  curve,  when 
rationalized  and  cleared  of  fractions,  the  lowest  terms  are  of 
degree  k,  the  origin  is  a  k-tuple  point  of  the  curve,  and  the  tan- 
gents at  this  point  are  given  by  the  terms  of  degree  k,  equated 
to  zero. 

To  investigate  whether  any  given  point  (xi ,  y^  of  an  alge- 
braic curve  is  simple  or  multiple  it  is  only  necessary  to  trans- 
fer the  origin  to  the  point,  by  replacing  xhy  x  +  x^^  and  y  by 
y  +  Vij  and  then  to  apply  this  rule. 

EXERCISES 

1.  Determine  the  nature  of  the  origin  and  sketch  the  curves  : 

{a)  y  =  x'^~-2x.        (b)  x^  =  4y-y\  (^c)  {x  +  a)(y  +  a)  =  a"^. 

(d)  ?/2  =  a;2(4-x).     {e)y^  =  3^.  (f)   x^  +  y'^  =  xK 

(g)  y^  =  x^  +  7?.         (h)  x^  -  3  axy  -\-y^  =  0.     (i)   x*-  y*  +  6  xy^  =  0. 

2.  Determine  the  nature  of  the  origin  and  sketch  the  curve  (y—x^y=x^, 
for:  (a)  n  =  l.         (6)  n  =  2.         (c)  w  =  3.        (d)  w  =  4. 

3.  Locate  the  multiple  points,  determine  their  nature,  and  sketch  the 
curves : 

(a)  y^  =  x{x  +  S)^.       (b)  (y-3)2  =  a;-2.       (c)  (y  ^  1)^  =  (x  -  S)\ 

(c?)y8=(x  +  l)(x-l)2. 

4.  Sketch  the  curve  y'^={x  —  a)(x  —  b){x—c)  and  discuss  the  multi- 
ple points  when : 

(a)  0<a<6<c.    (6)  0<a<&  =  c.    (c)  0<a  =  6<c.    {d)  0<a  =  b  =  c. 


254  PLANE  ANALYTIC  GEOMETRY      [XII,  §  266 

PART   11.     SPECIAL   CURVES 
DEFINED    GEOMETRICALLY   OR    KINEMATICALLY 

266.  Conchoid.  A  fixed  point  0  and  a  fixed  line  I,  at  the 
distance  a  from  O,  being  given,  the  radius  vector  OQ,  drawn  from 
0  to  every  point  Q  of  I,  is  produced  by  a  segment  QP=  b  of  con- 
stant length;  the  locus  of  P is  called  the  conchoid  of  Nicomedes. 

For  0  as  pole  and  the  perpendicular  to  I  as  polar  axis  the 
equation  of  lis  ri  =  a/  cos  <^ ;  hence  that  of  the  conchoid  is    -X 

If  the  segment  QP  be  laid  off  in  the  opposite  sense  we  obtain 
the  curve 

r  = b 

cos  <^ 

which  is  also  called  a  conchoid.  Indeed,  these  two  curves 
are  often  regarded  as  merely  two  branches  of  the  same 
curve.  Transforming  to  cartesian  coordinates  and  rationaliz- 
ing, we  find  the  equation 

(«-a)2(a;2-t-2/2)  =  6V, 

which  represents  both  branches.  Sketch  the  curve,  say  for 
b  =  2  a,  and  for  b  =  a/2,  and  determine  the  nature  of  the  origin. 

267.  Limacon.  If  the  line  I  be  replaced  by  a  circle  and  the 
fixed  point  0  be  taken  07i  the  circle,  the  locus  of  P  is  called 
Pascal's  limacon. 

For  0  as  pole  and  the  diameter  of  the  circle  as  polar  axis 

the  equation  of  the  circle,  of  radius  a,  is  r^  =  2  a  cos  <^ ;  hence 

that  of  the  limaqon  is  :  ^  ^^--^^ 

r  =  2  a  cos  <f>  -\- b.  t/y^ 


XII,  §  268] 


SPECIAL  CURVES 


255 


If   h  =  2a  the  curve   is   called  the  cardioid ;  the  equation 

then  becomes 

r  =  4  a  cos^  ^  4*. 

Sketch  the  limaqons  for  6  =  3  a,  2  a,  a ;  transform  to  car- 
tesian coordinates  and  determine  the  character  of  the  origin. 

268.  Cissoid.  00'  =  a  being  a  diameter  of  a  circle,  let  any 
radius  vector  drawn  from  0  meet  the  circle  and  its  tangent  at  0' 
at  the  points  Q,  D,  respectively;  if  on  this  radius  vector  we  lay 
off  OR  =  QD,  the  locus  of  E  is  called  the  cissoid  of  Diodes. 

With  0  as  pole  and  00'  as  polar  axis,  we  have 

OD  =  a/cos  <f),  OQ  =  a  cos  <^ ; 
the  equation  is  therefore 


=  fi( cos  A 1=  a 

\cos  <j>  J 


_^sin'<^^ 
cos  (^' 


or  in  cartesian  coordinates 
2  ^ 


Fig.  109 


If  instead  of  taking  the  difference  of  the  radii  vectores  of  the 
circle  and  its  tangent,  we  take  their  sum  we  obtain  the  so-called 
companion  of  the  cissoid, 

r  =  a(cos  <^  4-  sec  <^), 


I.e. 


Sketch  this  curve. 


2/2  =  x' 


2a  — X 
x  —  a 


IL  269.  Versiera.  With  the  data  of  §  268,  let  us  draw  through 
Q  a  parallel  to  the  tangent,  through  R  a  parallel  to  the  diameter ; 

■  the  locus  of  the  point  of  intersection  P  of  these  parallels  is 
called  the  versiera  (wrongly  called  the  "  witch  of  Agnesi "). 


256 


PLANE  ANALYTIC  GEOMETRY      [XII,  §  269 


We  have  evidently  with  0  as  origin  and  00'  as  axis  Ox 

x  =  a  cos^  <l>,        y  =  o.  tan  <^, 

whence  eliminating  <^ : 

a? 
x  = 

2/2  -+-  a2 

If  we  replace  the  tangent  at  0'  by  any 
perpendicular  to  00'  (Fig.  110),  at  the    ^ 
distance  h  from  0,  we  obtain  the  curve 
x  =  a  cos^  <^,        y  =  b  tan  <j>, 


which  reduces  to  the  versiera  for  b  =  a. 

Sketch  the  versiera,  and  the  last  curve  for  6  =  i  a. 


Fig.  110 


270.   Cassinian  Ovals.     Lemniscate.     Two  fixed  points  F„ 
F2  being  given  it  is  known  that  the  locus  of  a  point  P  is  : 


>    VK"^ 


Fig.  Ill 


(a)  a  circle  if  FiP/F.,P  =  const.  (Ex.  7,  p.  90); 
(6)  an  ellipse  if  F^P-^-F^P^z  const.  (§  204) ; 
(c)  a  hyperbola  if  i^iP- i?^2^=  const.  (§  207). 


The  locus  is  called  a  Cassinian  oval  if  JF\P  •  FgP  =  const.     If 


XII,  §  271] 


SPECIAL  CURVES 


257 


we  put  FiF2  =  2a,  the  equation,  referred  to  the  midpoint  0 
between  F^  and  F2  as  origin  and  OF2  as  axis  Ox,  is 

lix  +  af  4-  /]  [.{X  -  af  + 1/^  =  T^'- 

In  the  particular  case  when  k  =  a^  the  curve  passes  through 
the  origin  and  is  called  a  lemniscate.  The  equation  then  re- 
duces to  the  form 

{x^  +  y^y^2a\x^-y% 

which  becomes  in  polar  coordinates 

r"  =  2a2  cos  2  <^. 

Trace  the  lemniscate  from  the  last  equation. 

271.  Cycloid.  The  common  cycloid  is  the  path  described  by 
any  point  P  of  a  circle  rolling  over  a  straight  line  (Fig.  112). 


If  A  be  the  point  of  contact  of  the  rolling  circle  in  any  posi- 
tion, 0  the  point  of  the  given  line  that  coincided  with  the  point 
P  of  the  circle  when  P  was  point  of  contact,  it  is  clear  that 
the  length  OA  must  equal  the  arc  AP=a6,  where  a  is  the 
radius  of  the  circle,  and  6=  "^ACP  the  angle  through  which 
the  circle  has  turned  since  P  was  at  O.  The  figure  then  shows 
that,  with  O  as  origin  and  OA  as  axis  Ox : 

X  =  OQ  =  aO  —  a  sin  0,     y  =  a  —  a  cos  6. 

These  are  the  parameter  equations  of  the  cycloid.    The  curve  has 


258 


PLANE  ANALYTIC  GEOMETRY      [XII,  §  272 


an  infinite  number  of  equal  arches,  each  with  an  axis  of  sym- 
metry (in  Fig.  112,  the  line  x  =  ird)  and  with  a  cusp  at  each 
end.     Write  down  the  cartesian  equation. 

272.   Trochoid.     The  path  described  by  any  point  P  rigidly 
connected  with  the  rolling  circle  is  called  a  trochoid.     If  the 


Fig.  113.  —The  Trochoids 

distance  of  P  from  the  center  C  of  the  circle  is  6,  the  equations 
of  the  trochoid  are 

x=^ad  —  b  sin  $,    y  =  a  —  h  cos  6. 
Draw  the  trochoid  for  h  =  \a  and  f or  6  =  |  a. 


273.  Epicycloid.  The  path  described  by  any  point  P  of  a 
circle  rolling  on  the  outside  of  a  fixed  circle  is  called  an  epicy- 
cloid (Fig.  114). 

Let  0  be  the  center,  h  the 
radius,  of  the  fixed  circle,  C  the 
center,  a  the  radius,  of  the  rolling 
circle;  and  let  Aq  be  that  point 
of  the  fixed  circle  at  which  the 
describing  point  P  is  the  point 
of  contact.  Put  A^OA  =  <^,  ACP 
=  6.  As  the  arcs  AAq  and  AP 
are  equal,  we  have 

6<^  =  ad. 


Fig.  114 


XII,  §  274]  ^  SPECIAL  CURVES  259 

With  0  as  origin  and  OAq  as  axis  of  x  we  have 

a;  =  (a  H-  b)  cos  <f>  +  a  sin  [^  —  (|  rr  —  <^)], 
2/ =  (a  4- 6)  sin  <^  —  «  cos  [^  —  (|- TT  —  <^)], 

i.e,  x  =  (a-\-b)  cos  <^  —  a  cos  <^, 

2/  =  (a  +  o)  sm  <^  —  a  sin  — ! —  <f>. 

274.  Hypocycloid.  If  the  circle  rolls  on  the  inside  of  the 
fixed  circle,  the  path  of  any  point  of  the  rolling  circle  is  called 
a  hypocycloid.  The  equations  are  obtained  in  the  same  way ; 
they  differ  from  those  of  the  epicycloid  merely  in  having  a  re- 
placed by  —  a : 

X  =  (b  —  a)  cos  <f> -\-  a  cos    "~  ^  <^, 

CL 

y  =  (b  —  a)  sin  <^  —  a  sin     ~  ^  <^. 

Show  that :  (a)  for  b  =  2a  the  hypocycloid  reduces  to  a 
straight  line,  and  illustrate  this  graphically ;  (6)  for  b  =  4:a  the 
equations  become 

a;=  3  a  cos  <^-|-  a  cos  3  <f>  —  a  cos'  </>, 

2/ =  ^  <^  sin  <^  — a  sin  3  <^  =  asin' <^, 
whence  x^-\-y^  =  a^-^ 

sketch  this  four-cusped  hypocycloid. 

^  b  EXERCISES 

1.   Sketch  the  following  curves:    (a)   Spiral  of  Archimedes  r  =  a<f>;. 
(6)  Hyperbolic  spiral  r(^  a  ;  (c)  Lituus  r^^  =  a^.T^ 
.2.   Sketch  the  following  curves  :    (a)  r  =  a  sm<p  ;    (6)  r  =  a  cos  0  ; 
^cj)  r  =  a  sin  2  0  ;  (^r  =  a  cos  2  0  ;  (e)  r  =  a  cos  3  0  ;  (/)  r  =  a  sin  30  ; 
(g)  r  =  acos4  0;  (T^  r  =  a  sin  4  0. 

3.  Sketch  with  respect  to  the  same  axes  the  Cassinian  ovals  (§  270) 
for  a  =  1  and  k  =  2,  1.5,  1.1,  1,  .75,  .6,  0. 


260  PLANE  ANALYTIC  GEOMETRY      [XII,  §  274 

4.  Let  two  perpendicular  lines  AB  and  CD  intersect  at  0.  Through 
a  fixed  point  Q  of  AB  draw  any  line  intersecting  CD  at  E.  On  this  line 
lay  off  in  both  directions  from  B  segments  BP  of  length  OB.  The  locus 
of  P  is  called  the  strophoid.  Find  the  equation,  determine  the  nature  of 
O  and  Q,  and  sketch  the  curve. 

5.  Show  that  the  lemniscate  (§  270)  is  the  inverse  curve  of  an  equi- 
lateral hyperbola  with  respect  to  a  circle  about  its  center. 

6.  Show  that  the  strophoid  (Ex.  4)  is  the  curve  inverse  to  an  equilat- 
eral hyperbola  with  respect  to  a  circle  about  a  vertex  with  radius  equal 
to  the  transverse  axis. 

7.  Show  that  the  cissoid  (§  268)  is  the  curve  inverse  to  a  parabola 
with  respect  to  a  circle  about  its  vertex. 

8.  Find  the  curve  inverse  to  the  cardioid  (§267)  with  respect  to  a 
circle  about  the  origin. 

9.  Transform  the  equation 

a  (s;2  +  y2)  ^  r^z 

to  polar  coordinates,  indicate  a  geometrical  construction,  and  draw  the 
curve. 

10.  A  tangent  to  a  circle  of  radius  2  a  about  the  origin  intersects  the 
axes  at  T  and  2^,  find  and  sketch  the  locus  of  the  midpoint  P  between  T 
and  T'. 

11.  From  any  point  Q  of  the  line  x  =  a  draw  a  line  parallel  to  the  axis 
Ox  intersecting  the  axis  Oy  at  C  Find  and  sketch  the  locus  of  the  foot 
of  the  perpendicular  from  O  on  OQ. 

12.  The  center  of  a  circle  of  radius  a  moves  along  the  axis  Ox.  Find 
and  sketch  the  locus  of  the  intersections  of  this  circle  with  lines  joining 
the  origin  to  its  highest  point. 

13.  The  center  of  a  circle  of  radius  a  moves  along  the  axis  Ox.  Find 
and  sketch  the  locus  of  its  points  of  contact  with  the  lines  through  the  origin. 

14.  The  center  of  a  circle  of  radius  a  moves  along  the  axis  Ox.  Its  in- 
tersection with  the  axis  nearer  the  origin  is  taken  as 'the  center  of  another 
circle  which  passes  through  the  origin.  Find  and  sketch  the  locus  of  the 
intersections  of  these  circles. 


XII,  §  276]         TRANSCENDENTAL  CURVES  261 

PART  III.     SPECIAL  TRANSCENDENTAL  CURVES 

275.  The  Sine  Curve.  The  simple  sine  curve,  y  =  sin  x, 
is  best  constructed  by  means  of  an  auxiliary  circle  of  radius 
one.  In  Fig.  115,  OQ  is  made  equal  to  the  length  of  the  arc 
OA  =  X ;  the  ordinate  at  Q  is  then  equal  to  the  ordinate  BA  of 
the  circle. 

y 


Fig.  115 

Construct  one  whole  j^enod  of  the  sine  curve,  i.e.  the  portion 
corresponding  to  the  whole  circumference  of  the  auxiliary 
circle ;  the  width  2  ^  of  this  portion  is  called  the  period  of  the 
function  sinx. 

The  simple  cosine  curve,  y  =  cos  x,  is  the  same  as  the  sine 
curve  except  that  the  origin  is  taken  at  the  point  (^tt,  0). 

The  simple  tangent  curve,  y  =  tan  x,  is  derived  like  the  sine 
curve  from  a  unit  circle.     Its  period  is  tt. 

276.  The  Inverse  Trigonometric  Curves.  The  equation 
y  =  sin  a;  can  also  be  written  in  the  form 

X  =  sin~^  y,     or  a;  =  arc  sin  y. 
The  curve  represented  by  this  equation  is  of  course  the  same 
as  that  represented  by  the  equation  y  =  sin  x. 

But  if  X  and  y  be  interchanged,  the  resulting  equation 
X  =  sin  y,  or  y  =  sin~^  x,  y  —  arc  sin  x, 
represents  the  curve  obtained  from  the  simple  sine  curve  by 
reflection  in  the  line  y=x(^  135). 


262  PLANE  ANALYTIC  GEOMETRY      [XII,  §  276 

Notice  that  the  trigonometric  functions  sin  x,  cos  x,  tan  x,  etc., 
^re  one-valued,  i.e.  to  every  value  of  x  belongs  only  one  value 
of  the  function,  while  the  inverse  trigonometric  functions  sin~^  a;, 
cos~^a7,  tan~^a?,  etc.,  are  many-valued;  indeed,  to  every  value  of 
X,  at  least  in  a  certain  interval,  belongs  an  infinite  number  of 
values  of  the  function. 

EXERCISES 

1.  From  a  table  of  trigonometric  functions,  plot  the  curve  y  =  sinx. 

2.  Plot  the  curve  y  =  sinx  by  means  of  the  geometric  construction 
of  §275. 

3  Plot  the  curve  y  =  cosx  (a)  from  a  table ;  (6)  by  a  geometric  con- 
struction similar  to  that  of  §  275. 

4.  Plot  the  curve  y  =  tan  x  from  a  table. 

5.  Plot  each  of  the  curves 

(a)  y  =  sm2  x.  (d)  y  =  sec  x. 

(6)  y  =  2  cos  3  a;.  {e)  y  =  ctn  2  x. 

(c)  y  =  3  tan  (x/2).  (/)  y  =  2  tan  4  x. 

6.  Plot  each  of  the  curves 

(a)  y  =  sin-i  x.  (6)  y  =  cos-i  x.  (c)  y  =  tan-i  x. 

7.  By  adding  the  ordinates  of  the  tw^o  curves  y  =  sin  x  and  y  =  cos  x, 
construct  the  graph  oi  y  =  sin  x  +  cos  x. 

8.  Draw  each  of  the  curves 

(a)  y  zzsiux  +  2  cos  x.  (c)  y  =  secx-{-  tan  x. 

(6)  y  =  2  sin  x  +  cos(x/2).  (d)  ?/  =  sin  x  +  2  sin  2  x  +  3  sin  3x. 

9.  The  equation  x  =  sin  t,  where  t  means  the  time  and  x  means  the 
distance  of  a  body  from  its  central  position,  represents  a  Simple  Harmonic 
Motion.  From  the  graph  of  this  equation,  describe  the  nature  of  the 
motion. 

277.  Transcendental  Curves.  The  trigonometric  and  in- 
verse trigonometric  curves,  as  well  as,  in  general,  the  cycloids 
and  trochoids,  are  transcendental  curves,  so  called  because  the 
relation  between  the  cartesian  coordinates  x,  y  cannot  be  ex- 
pressed in  finite  form  {i.e.  without  using  infinite  series)  by 


XII,  §  279]         TRANSCENDENTAL  CURVES 


263 


means  of  the  algebraic  operations  of  addition,  subtraction,  mul- 
tiplication, division,  and  raising  to  a  power  with  a  constant 
exponent. 

278.  Logarithmic  and  Exponential  Curves.  Another  very 
important  transcendental  curve  is  the  exponential  curve 

y  =  «^ 

and  its  inverse,  the  logarithmic 

curve  1 

y  =  log«  X, 

where  a  is  any  positive  constant 
(Fig.  116).  A  full  discussion 
of  these  curves  can  only  be  given 
in  the  calculus.  We  must  here 
confine  ourselves  to  some  special 
cases  and  to  a  brief  review  of  the 
fundamental  laws  of  logarithms. 

279.  Definitions.  The  logarithm  6  of  a  number  c,  to  the 
base  a  (positive  and  ^  1),  is  defined  as  the  exponent  b  to  which 
the  given  base  a  must  be  raised  to  produce  the  number  c 
(§  105)  ;  thus  the  two  equations 

a^  =  c  and  b  =  log„  c 

express  exactly  the  same  relation  between  b  and  c.  It  follows 
that  a'"""''  =  c,  whatever  c. 

If  in  the  first  law  of  exponents  (§  104),  a^a'' =  a^'^'^,  we  put 
aP=Py  a«=  Q,  a'+«=iV,  so  that  PQ=N,  we  find  since  p=loga  P, 

q  =  loga  Q,p  +  q  =  loga  ]sr=  log,  pq  -. 

(1)  log„PQ  =  log,P+log„Q. 

Similarly  we  find  from  a^/a*^  =  a^'" : 
P 


(2) 


loga  ^=  log.  P-l0g„Q. 


264  PLANE  ANALYTIC  GEOMETRY      [XII,  §  279 

If  in  the  third  law  of  exponents  (§  104),  (a^y  =  a^",  we  put 
.a**  =  P,  aP"  =  M,  so   that   P""  =  M,  we   find   since  p  =  log„  P, 
pn  =  log„  M: 
(3)  log„(P")  =  nlog,P. 

These  laws  (1),  (2),  (3)  of  logarithms  are  merely  the  trans- 
lation into  the  language  of  logarithms  of  the  first  and  third 
laws  of  exponents. 

280.  Napierian  or  Natural  Logarithms.  In  the  ordinary 
tables  of  logarithms  the  base  is  10,  and  for  numerical  calcula- 
tions these  common  logarithms  (Briggs'  logarithms)  are  most 
■convenient.  In  the  calculus  it  is  found  that  another  system 
of  logarithms  is  better  adapted  to  theoretical  considerations ; 
the  base  of  this  system  is  an  irrational  number  denoted  by  e, 

6  =  2.718281828  ..., 

and  the  logarithms  in  this  system  are  called  natural  logarithms 
(or  Napierian,  or  hyperbolic,  logarithms). 

281.  Change  of  Base.  Modulus.  To  pass  from  one  system 
of  logarithms  to  another  observe  that  if  the  same  number  N  has 
the  logarithm  p  in  the  system  to  the  base  a  and  the  logarithm 
g  in  the  system  to  the  base  b  so  that 

a^  =  N,  p=  log„  N,  h"  =  N,  q=  log,  N, 
then  q  =  logj,  N=  log^  a^=p  log^,  a, 

by  (3);  i.e. 
,(4)  iogi^=log„^.logj,a. 

Hence  if  the  logarithms  of  the  system  with  the  base  a  are 
known,  those  with  the  base  b  are  found  by  multiplying  the 
logarithms  to  the  base  a  by  a  constant  number,  logj,a. 
Thus  taking  a  =  10,  b  =  e,  we  have 

(4')  log,iV^=log,o^^.log,10; 


XII,  §281]        TRANSCENDENTAL  CURVES  265 

i.e.  to  find  the  natural  logarithm  of  any  number  we  have  merely 
to  multiply  its  common  logarithm  by  the  number 

log,  10  =  2.30258  509  .... 
The  reciprocal  of  this  number, 

M=  — i—  =  0.43429  448  •  • ., 
logg  10 

i.e.  the  factor  by  which  the  natural  logarithms  must  be  multi- 
plied to  produce  the  common  logarithms,  is  called  the  modulus 
of  the  common  system  of  logarithms. 

In  any  system  of  logarithms,  the  logarithm  of  the  base  is 
always  equal  to  1,  by  the  definition  of  the  logarithm  (§  279). 
Hence,  if  in  (4)  we  take  iV"=  &,  we  find 
(5)  log„6  .log,a  =  l. 

In  particular,  with  a  =  10,  6  =  e  we  have 
(5')  log.oe.  log,  10  =  1; 

i.e.  the  modulus  M  of  the  common  logarithms  is 

Jf  = —i— =  logio  e  =  0.43429  448  ...  . 
log,  10 

EXERCISES 

1.  From  a  table  of  logarithms  of  numbers,  draw  the  curve  y  =  logio  x. 

2.  By  multiplying  the  ordinates  of  the  curve  of  Ex.  1  by  3,  construct 
the  curve  y  =  S  logio  x. 

3.  From  the  figure  of  Ex.  1,  construct  the  curve  y  =  10*  by  reflection 
of  the  curve  of  Ex.  1  in  the  line  y  =  x. 

4.  Draw  the  curve  y  =  ^  logio  x  by  the  process  of  Ex.  2.     Show  that  it 
represents  the  equation  y  =  logioo  x,  since 

y  =  logioo  X  =  logioo  10  X  logic  x  =  ^  logic  X. 

5.  Find  logio  7  from  a  table.     Construct  the  curve 

y  =  logy  X  =  logioa;  ^  logio  7 
by  the  process  described  in  Ex.  2  and  Ex.  4. 

6.  Given  logic  e  =  M=  .43+,  draw  the  curve 

y  =  log,  X  =  logic  X  ^  logic  e. 


266  PLANE  ANALYTIC  GEOMETRY      [XII,  §  282 

PART  IV.     EMPIRICAL  EQUATIONS 

282.  Empirical  Formulas.  In  scientific  studies,  the  rela- 
tions between  quantities  are  usually  not  known  in  advance, 
but  are  to  be  found,  if  possible,  from  pairs  of  numerical  values 
of  the  quantities  discovered  by  experiment. 

Simple  cases  of  this  kind  have  already  been  given  in  §§  15, 
29.  In  particular,  the  values  of  a  and  h  in  formulas  of  the 
type  y  =  a-{-bx  were  found  from  two  pairs  of  values  of  x  and  y. 
Compare  also  §  34. 

Likewise,  if  two  quantities  y  and  x  are  known  to  be  connected 
by  a  relation  of  the  form  y  =  a-\-bx-\-  cx"^,  the  values  of  a,  b,  c 
can  be  found  from  any  three  pairs  of  values  of  x  and  y.  For, 
if  any  pair  of  values  of  x  and  y  are  substituted  for  x  and  y 
in  this  equation,  we  obtain  a  linear  equation  for  a,  b,  and  c. 
Three  such  equations  usually  determine  a,  b,  and  c. 

In  general  the  coefficients  a,  b,  c,  •••,  Z  in  an  equation  of  the 

^^  y  z=  a -\- bx -\- cx"^ -\-  " '  -\- Ix"" 

can  be  found  from  any  n  +  1  pairs  of  values  of  x  and  y. 

283.  Approximate  Nature  of  Results.  Since  the  measure- 
ments made  in  any  experiment  are  liable  to  at  least  small 
errors,  it  is  not  to  be  expected  that  the  calculated  values  of 
such  coefficients  as  a,  &,  c,  •  •  •  of  §  282  will  be  absolutely  accu- 
rate, nor  that  the  points  that  represent  the  pairs  of  values  of 
X  and  y  will  all  lie  absolutely  on  the  curve  represented  by  the 
final  formula. 

'  To  increase  the  accuracy,  a  large  number  of  pairs  of  values 
of  X  and  y  are  usually  measured  experimentally,  and  various 
pairs  are  used  to  determine  such  constants  as  a,  &,  c,  --of  §  282. 
The  average  of  all  the  computed  values  of  any  one  such  con- 
stant is  often  taken  as  a  fair  approximation  to  its  true  value. 


XII,  §  284] 


EMPIRICAL  EQUATIONS 


267 


284.   Illustrative  Examples. 

Example  1.  A  wire  under  tension  is  found  by  experiment  to  stretch 
an  amount  I,  in  thousandths  of  an  inch,  under  a  tension  T,  in  pounds,  as 
follows :  — 

T  in  pounds 10  15  20  25  30 

I  in  thousandths  of  an  inch  .        8  12.5         15.5  20  23 

Find  a  relation  of  the  form  I  =  kT  (Hookers  Law)  which  approx- 
imately represents  these  results. 

First  plot  the  given  data  on  squared  paper,  as  in  the  adjoining  figure. 


dU 

'~' 

— 

— 

"" 

""■ 

"~* 

■"" 

■"" 

25 

/ 

/ 

< 

./ 

/ 

/ 

20 

^ 

/ 

/ 

/' 

/ 

/ 

15 

t 

V( 

) 

,/ 

/ 

y 

/ 

10 

,/ 

/ 

/ 

J 

/ 

^/ 

^ 

5 

/ 

/ 

/ 

/ 

/ 

/ 

0 

\ 

s 

\ 

n 

\ 

"i 

7 

0 

2 

5 

.-"i 

n 

7s^ 

Fig.  117 

Substituting  ?  =  8,  T  =  10  in  ?  =  A:r,  we  find  A;  =  .8.  From  I  =  12.5, 
T  —  15,  we  find  k  =  .833.  Likewise,  the  other  pairs  of  values  of  I  and  T 
give,  respectively,  k  =  .775,  k  =  .8,  k=  .767.  The  average  of  all  these 
values  of  A;  is  A:  =  .795  ;  hence  we  may  write,  approximately, 

I  =  .795  T. 


268 


PLANE  ANALYTIC  GEOMETRY      [XII,  §  284 


This  equation  is  represented  by  the  line  in  Fig.  117  ;  this  line  does  not 
pass  through  even  one  of  the  given  points,  but  it  is  a  fair  compromise  be- 
tween all  of  them,  in  view  of  the  fact  that  each  of  them  is  itself  probably 
slightly  inaccurate. 

Example  2.  In  an  experiment  with  a  Weston  Differential  Pulley 
Block,  the  effort  E^  in  pounds,  required  to  raise  a  load  IF,  in  pounds,  was 
found  to  be  as  follows  : 


w 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

E 

3i 

4| 

6i 

n 

9 

101 

12i 

13f 

15 

161 

Find  a  relation  of  the  form  E 
with  these  data. 


aW  +h  that  approximately  agrees 
[Gibson] 


These  values  may  be  plotted  in  the  usual  manner  on  squared  paper. 
They  will  be  found  to  lie  very  -^ 
nearly  on  a  straight  line.  If  E 
is  plotted  vertically,  h  is  the  in- 
tercept on  the  vertical  axis,  and 
a  is  the  slope  of  the  line  ;  both 
can  be  measured  directly  in  the 
figure. 

To  determine  a  and  h  more 
exactly,  we  may  take  various 
points  that  lie  nearly  on  the 
line.  Thus  {E  =  Q\,  pr=30) 
and  {E  =  16^,  W  =  100)  lie 
nearly  on  a  line  that  passes  close 
to  all  the  points.    Substituting  in  the  equation  E  =  aW  -^  h  ^e  obtain 

6|  =  30a  +  6,        16J  =  100a+& 

whence  a  =  0.146,     h  =  1.86.     Hence  we  may  take 

E=  0.146  Tr+  1.86 

approximately.  Other  pairs  of  values  of  E  and  W  may  be  used  in  like 
manner  to  find  values  f  or  «  and  6,  and  all  the  values  of  each  quantity  may 
be  averaged. 


T 

:                  ^' 

^«  - 

1^ 

»^ 

.^ 

_       1 

i>  r 

10         -        -                 - 

V      ~~      ~  " 

^' 

-^  ^ 

s*  -  - 

5         -     -jr---        -     - 

a^ 

y' 

'.                        w 

V               20            40 

60           80          IDO. 

Fig.  118 


XII,  §284]  EMPIRICAL  EQUATIONS  269 

Example  3.  If  6  denotes  the  melting  point  (Centigrade)  of  an  alloy 
of  lead  and  zinc  containing  x  per  cent  of  lead,  it  is  found  that 

X  =  %  lead 40         50         60         70         80         90 

^  =  melting  point     ....     186°      205°      226°      250°      276°      304° 

Find  a  relation  of  the  form  6  =  a  -{•  bx  +  cx^  that  approximately  expresses 
these  facts.  [Saxelby] 

Taking  any  three  pairs  of  values,  say  (40,  186),  (70,  250),  (90,  304), 
and  substituting  in  d  =  a  -\- bx -]-  cx^  we  find 

186  =  «  +  40  6  +  1600  c, 

260=  a +  70  b  +  4900  c, 

304  =  a  +  90  6  +  8100  c, 

whence  a  =  132,  b  =  .92,  c  =  .0011,  approximately  ;  whence 

e  =  132  +  .92x+  .0011x2. 

Other  sets  of  three  pairs  of  values  of  x  and  y  may  be  used  in  a  similar 

manner  to  determine  «,  6,  c ;  and  the  resulting  values  averaged,  as  above. 

EXERCISES 

1.  In  experiments  on  an  iron  rod,  the  amount  of  elongation  I  (in  thou- 
sandths of  an  inch)  and  the  stretching  force  p  (in  thousands  of  pounds) 
were  found  to  be  {p  =  10,  l=S),  (p  =  20,  Z  =  15),  (p  =  40,  Z  =  31). 
Find  a  formula  of  the.  type  l=k-p  which  approximately  expresses  these 
data.  Ans.  k  =  .775. 

2.  The  values  1  in.  =2.5  cm.  and  1  ft.  =30.5  cm.  are  frequently 
quoted,  but  they  do  not  agree  precisely.  The  number  of  centimeters,  c, 
in  i  inches  is  surely  given  by  a  formula  of  the  type  c  =  ki.  Find  k  ap- 
proximately from  the  preceding  data. 

3.  The  readings  of  a  standard  gas-meter  S  and  those  of  a  meter  T  being 
tested  on  the  same  pipe-line  were  found  to  be  (<S'=3000,  r=0),  (*9=3510, 
T  =  500),  (S  =  4022,  T  =  1000) .  Find  a  formula  of  the  type  T=  aS+  b 
which  approximately  represents  these  data. 

4.  An  alloy  of  tin  and  lead  containing  x  per  cent  of  lead  melts  at  the 
temperature  d  (Fahrenheit)  given  by  the  values  (a:  =  25%,  ^  =  482°), 
(x  =  50%,  d  =  370°),  (ic  =  75%,  d  =  356°).  Determine  a  formula  of  the 
type  6  =  a  +  bx  +  cx^  which  approximately  represents  these  values. 


270  PLANE  ANALYTIC  GEOMETRY      [XII,  §  284 

5.  The  temperatures  d  (Centigrade)  at  a  depth  d  (feet)  below  the  sur- 
face of  the  earth  in  a  mine  were  found  to  be  <?  =  100,  6  =  15.7°  ;  d  =  200, 
^=16.5 ;  d=300,  ^=17.4.  Find  a  relation  of  the  form  d=a-\-bd  between 
e  and  d. 

6.  Determine  a  line  that  passes  reasonably  near  each  of  the  three 
points  !(2,  4),  (6,  7),  (10,  9).  Determine  a  quadratic  expression 
y=a  +  hx-\-cx^  that  represents  a  parabola  through  the  same  three  points. 

7.  Determine  a  parabola  whose  equation  is  of  the  form  y  =  a-}-bx-\-cx^ 
that  passes  through  each  of  the  points  (0,  2.5),  (1.5,  1.5),  and  (3.0,  2.8). 
Are  the  values  of  «,  &,  c  changed  materially  if  the  point  (2.0,  1.7)  is 
substituted  for  the  point  (1.5,  1.5)  ? 

8.  If  the  curve  y  =  sinx  is  drawn  with  one  unit  space  on  the  ic-axis 
representing  60^,  the  points  (0,  0),  (^,  J),  (I2,  1)  lie  on  the  curve.  Find  a 
parabola  of  the  form  y=a-\-bx-{-cx^  through  these  three  points,  and  draw 
the  two  curves  on  the  same  sheet  of  paper  to  compare  them. 

285.  Substitutions.  It  is  particularly  easy  to  test  whether 
points  that  are  given  by  an  experiment  really  lie  on  a  straight 
line ;  that  is,  whether  the  quantities  measured  satisfy  an  equa- 
tion of  the  form  y  =  a-\-bx.  This  is  done  by  means  of  a  trans- 
parent ruler  or  a  stretched  rubber  band. 

For  this  reason,  if  it  is  suspected  that  two  quantities  x  and 
y  satisfy  an  equation  of  the  form 

y  =  a  +  bx\ 
it  is  advantageous  to  substitute  a  new  letter,  say  u,  for  x^ : 

u  =  x^j     y  =  a  -{-  bu 
and  then  plot  the  values  of  y  and  u.     If  the  new  figure  does 
agree  reasonably  well  with  some  straight  line,  it  is  easy  to  find 
a  and  6,  as  in  §  284. 

Likewise,  if  it  is  suspected  that  two  quantities  x  and  y  are 
connected  by  a  relation  of  the  form 

2/  =  a  -f  6  •  -  or  xy  =  ax-{-bf 

X 

it  is  advantageous  to  make  the  substitution  u  =  1/x. 


XII,  §  286]  EMPIRICAL  EQUATIONS  271 

Other  substitutions  of  the  same  general  nature  are  often 
useful. 

In  any  case,  the  given  values  of  x  and  y  should  he  plotted  first 
unchanged,  in  order  to  see  what  substitution  might  he  useful, 

286.  Illustrative  Example.  If  a  body  slides  down  an  inclined 
plane,  the  distance  s  that  it  moves  is  connected  with  the  time  t  after  it 
starts  by  an  equation  of  the  form  s  =  kP:  Find  a  value  of  k  that  agrees 
reasonably  with  the  following  data  : 

s,  in  feet 2.6  10.1  23.0  40.8  63.T 

t,  in  seconds 1  2  3  4  5 

In  this  case,  it  is  not  necessary  to  plot  the  values  of  s  and  t  themselves, 
because  the  nature  of  the  equation,  s  =  kt'^^  is  known  from  physics. 

Hence  we  make  the  substitution  t^  =  u,  and  write  down  the  supple- 
mentary table : 

s,  in  feet 2.6  10.1  23.0  40.8  63.7 

w  (or  «2) 1  4  9  16  25 

These  values  will  be  found  to  give  points  very  nearly  on  a  straight  line 
whose  equation  is  of  the  form  s  =  ku.  To  find  k,  we  divide  each  value  of 
s  by  the  corresponding  value  of  u  ;  this  gives  several  values  of  k : 

k  2.6        2.525        2.556         2.55         2.548 

The  average  of  these  values  of  k  is  approximately  2.556  ;  hence  we  may 
write  s  =  2.556  m,  or  s  =  2.556  t^. 

EXERCISES 

1.  Find  a  formula  of  the  type  u  =  kv^  that  represents  approximately 
the  following  values  :  " 

tt  3.9  15.1  34.5  61.2  95.5         137.7        187.4 

t;12  34567 


272  PLANE  ANALYTIC  GEOMETRY      [XII,  §  286 

2.  A  body  starts  from  rest  and  moves  s  feet  in  t  seconds  according  to 
the  following  measured  values  : 

s,  in  feet 3.1         13.0        30.6        50.1        79.5       116.4 

t,  in  seconds 5  1  15  2  2.5  3 

Find  approximately  the  relation  between  s  and  t. 

3.  The  pressure  p,  measured  in  centimeters  of  mercury,  and  the  volume 
V,  measured  in  cubic  centimeters,  of  a  gas  kept  at  constant  temperature, 
were  found  to  be  : 


145 

155 

165 

178 

191 

L17.2 

109.4 

102.4 

95.0 

88.6 

p 

Substitute  u  for  l/u,  compute  the  values  of  m,  and  determine  a  relation 
of  the  form  p  =  ku;  that  is,  p  =  k/v. 

4.   Determine  a  relation  of  the  form  y  =  a  +  bx^  that  approximately 
represents  the  values : 

X  1  2  3  4  5  6  7 

y  14.1  25.2         44.7         71.4         105.6        147.9        197.7 

287.   Logarithmic  Plotting.      In  case  the  quantities  y  and  x 
are  connected  by  a  relation  of  the  form 

y  =  kx% 

it  is  advantageous  to  take  logarithms  (to  the  base  10)  on  both 

sides : 

log  y  =  log  Tex""  =  log  k  -{-n  log  x, 

and  then  substitute  new  letters  for  log  x  and  log  y : 

u  =  log  Xj        V  —  log  y. 

For,  if  we  do  so,  the  equation  becomes 

v  =  l  -{-  nu, 
where  I  =  log  k. 


XII,  §  287] 


EMPIRICAL  EQUATIONS 


273 


If  the  values  of  x  and  y  are  given  by  an  experiment,  and  if 
u  =  log  X  and  v  =  log  y  are  computed,  the  values  of  u  and  v 
should  correspond  to  points  that  lie  on  a  straight  line,  and  the 
values  of  I  and  n  can  be  found  as  in  §  284.  The  value  of  k 
may  be  found  from  that  of  /,  since  log k=  l. 

Example  1.  The  amount  of  water  A,  in  cu.  ft.  that  will  flow  per 
minute  through  100  feet  of  pipe  of  diameter  d,  in  inches,  with  an  initial 
pressure  of  50  lb.  per  sq.  in.,  is  as  follows  : 


d 

1 

1.5 

2 

3 

4 

6 

A] 

4.88 

13.43 

27.50 

75.13 

152.51 

409.54 

Fmd  a  relation  between  A  and  d. 


Let  u  =  \ogd,  V  =  log  A  ;  then  the  values  of  u  and  v  are 


u  =  \ogd  .     . 

.      0.000 

0.176 

0.301 

0.477 

0.602 

0.778 

v  =  \ogA.     . 

.      0.688 

1.128 

1.439 

1.876 

2.183 

2.612 

::::::::::::         :::::::::::-5=::::::: 


i  .2  .3  4  .5 

Fig.  119 


j6         .7        .8 


These  values  give  points  in  the  (u,  v)  plane  that  are  very  nearly  on 
a  straight  line  ;  hence  we  may  write,  approximately, 

V  =  a+  bu, 

where  a  and  b  can  be  determined  directly  by  measurement  in  the  figure, 

T 


274  PLANE  ANALYTIC  GEOMETRY      [XII,  §  287 

or  as  in  §  284.     If  we  take  the  first  and  last  pairs  of  values  of  u  and  v,  we 

find 

.688  =  a  +  0, 

2.612  =  a  +  .778&. 

Solving  these  equations,  we  find  approximately,  a  =  .688,  b  =  2.473, 
and  we  may  vvrrite 

V  =  .688  +  2.473  u    or    log  A  =  .688  +  2.473  log  d. 

Since  .688  =  log  4.88, 

the  last  equation  may  be  wrritten  in  the  form 

log  A  =  log  4.88  +  2.473  log  d 

=  log(4.88c?2-*78) 

whence  ^  =  4.88  (?2. 473. 

Slightly  different  values  of  the  constants  may  be  found  by  using  other 
pairs  of  values  of  u  and  v. 

288.  Logarithmic  Paper.  Paper  called  logarithmic  paper 
may  be  bought  that  is  ruled  in  lines  whose  distances,  horizon- 
tally and  vertically,  from  one  point  0  (Fig.  120)  are  propor- 
tional to  the  logarithms  of  the  numbers  1,  2,  3,  etc. 

Such  paper  may  be  used  advantageously  instead  of  actually 
looking  up  the  logarithms  in  a  table,  as  was  done  in  §  287. 
For  if  the  given  values  be  plotted  on  this  new  paper,  the  result- 
ing figure  is  identically  the  same  as  that  obtained  by  plotting 
the  logarithms  of  the  given  values  on  ordinary  squared  paper. 

Example.  A  strong  rubber  band  stretched  under  a  pull  of  p  kg. 
shows  an  elongation  of  E  cm.  The  following  values  were  found  in  an  ex- 
periment : 

p        0.5      1.0     1.5     2.0     2.5     3.0     3.5     4.0     4.5     6.0     6.0     7.0 
E        0.1      0.3     0.6     0.9     1.3     1.7     2.2     2.7     3.3     3.9     5.3     6.9 

[RiGGS] 

If  these  values  are  plotted  on  logarithmic  paper  as  in  Fig.  120,  it  is  evi- 
dent that  they  lie  reasonably  near  a  straight  line,  such  as  that  drawn. 


XII,  §  288] 


EMPIRICAL  EQUATIONS 


275 


By  measurement  in  the  figure,  the  slope  of  this  line  is  found  to  be  1.6 
approximately.     Hence  if  m  =  log j?  and  v  =  log  ^  we  have 

where  I  is  a  constant  not  yet  determined  ;  whence 

log^=:Z  +  1.6  1ogp 
or  E  =  A.pi-6, 


<n^ 

ltf_... 

7: 

. 

t 

-J-  - 

;' 

^ 

/ 

/ 

/ 

"  E  =  elongation  in  c 
p  =  pull  in  kg. 

/ 

3 

m. 

J 

2 

'  Ez=.2 

pi 

.6 

— 

15 

= 

^ 

— _ 

! 

' 

/ 

2-:_ 

:::::i;i 

:       z_ 

f' 

T    J 

) 

fc    _ 

/ 

_l 

z 

•z    __ 

k  =  .3;z  '^ 

7 

.2---- 

= 

7"~~ 

= 

15 

E 

z2 

Trrrrr 

E 

X 

__ 

.^L 

i  iB      .2  .5      .4    .5  .6  .7.8.91  15      2  "S       4     5    6  7  8  910' 

Fig.  120.  — Elongation  of  a  Rubber  Band 

where  Z  =  log  A;.     If  j?  =  1,  JS'  =  A: ;  from  the  figure,  if  p  =  1,  ^  =  .3  ; 
hence  A:  =  .3,  and 

E  =  .3j!)l-6. 

The  use  of  logarithmic  paper  is  however  not  at  all  essential ; 
the  same  results  may  be  obtained  by  the  method  of  §  287. 


276  PLANE  ANALYTIC   GEOMETRY       [XII,  §  288 

EXERCISES 

1.  In  testing  a  gas  engine  corresponding  values  of  the  pressure  p^  meas- 
ured in  pounds  per  square  foot,  and  the  volume  v,  in  cubic  feet,  were 
obtained  as  follows  :  v  =  7.14,  p  =  54.6  ;  7.73,  50.7  ;  8.59,  45.9.  Find 
the  relation  between p  and  v  (use  logarithmic  plotting). 

Ans.  p  =  387.6  v-^^,  or  pv^  =  387.6. 

2.  Expansion  or  contraction  of  a  gas  is  said  to  be  adiabatic  when  no 

heat  escapes  or  enters.   Determine  the  adiabatic  relation  between  pressure 

p  and  volume  v  (Ex.   14)  for  air  from  the  following  observed  values : 

p  =  20.54,  V  =  6.27  ;  25.79,  5.34  ;  54.25,  3.15. 

Ans.  pv^-'^  =  273.5. 

3.  The  intercollegiate  track  records  for  foot-races  are  as  follows, 
where  d  means  the  distance  run,  and  t  means  the  record  time  : 

d  100  yd.      220  yd.      440  yd.      880  yd.        1  mi.  2  mi. 

t  0:09|  0:21^  0:48  l:54f  4:15f  9:24| 

Plot  the  logarithms  of  these  values  on  squared  paper  (or  plot  the 
given  values  themselves  on  logarithmic  paper).      Find  a  relation  of  the 
form  t  =  kd\     What  should  be  the  record  time  for  a  race  of  1320  yd.  ? 
[See  Kennelly,  Popular  Science  Monthly,  Nov.  1908.] 

4.  Solve  the  Example  of  §  288  by  the  method  of  §  287. 

5.  Each  of  the  following  sets  of  quantities  was  found  by  experiment. 
Find  in  each  case  an  equation  connecting  the  two  quantities,  by  §§  287- 
288. 


(a)   V 
P 

1 

137.4 

2 
62.6 

3 

39.6 

4 

28.6 

5 
22.6 

(6)  u 

V 

12.9 
63.0 

17.1 
27.0 

23.1 
13.8 

28.5 
8.5 

3.0 

6.9 

(c)  e 

c 

82^^ 
2.09 

212° 
2.69 

390° 
2.90 

570° 
2.98 

750° 
3.09 

1100^ 
3.28 

SOLID    ANALYTIC    GEOMETRY 


CHAPTER   XIII 


COORDINATES 

289.  Location  of  a  Point.  The  position  of  a  point  in  three- 
dimensional  space  can  be  assigned  without  ambiguity  by  giv- 
ing its  distances  from  three  mutually  rectangular  planes,  pro- 
vided these  distances  are  taken  with  proper  signs  according  as 
the  point  lies  on  one  or  the  other  side  of  each  plane. 

The  three  planes,  each  perpendicular  to  the  other  two,  are 
called  the  coordinate  planes ;  their  common  point  0  (Fig.  121) 
is  called  the  origin.  The  three 
mutually  rectangular  lines  Ox, 
Oy,  Oz  in  which  the  planes  in- 
tersect are  called  the  axes  of 
coordinates;  on  each  of  them 
a  positive  sense  is  selected 
arbitrarily,  by  affixing  the 
letter  x,  y,  z,  respectively. 

The  three  coordinate  planes, 
Oyz,  Ozx,  Oxy,  divide  the  whole 
of  space  into  eight  compartments  called  octants.  The  first 
octant  in  which  all  three  coordinates  are  positive  is  also  called 
the  coordinate  trihedral. 

If  P',  P",  P'"  are  the  projections  of  any  point  P  on  the 
coordinate  planes  Oyz,  Ozx,  Oxy,  respectively,  then  P'P=x, 
P"P  =  y,  P'"P=  z  are  the  rectangular  cartesian  coordinates  of 

277 


/! 

/ 

1 

^y 

z 

?/"-- 

• 

""-^ 

V 

Q'         y 

Fig.  121 


278 


SOLID  ANALYTIC  GEOMETRY      [XIII,  §  289 


P.  If  the  planes  through  P  parallel  to  Oyz,  Ozx,  Oxy  intersect 
the  axes  Ox,  Oy,  Oz  in  Q',  Q",  Q'",  the  point  P  is  found  from 
its  coordinates  x,  y,  z  by  passing  along  the  axis  Ox  through  the 
distance  0Q'=  x,  parallel  to  Oy  through  the  distance  Q'P"=yj 
and  parallel  to  Oz  through  the  distance  P"P=z,  each  of 
these  distances  being  taken  with  the  proper  sense. 

Every  point  in  space  has  three  definite  real  numbers  as  coordi- 
nates; conversely,  to  every  set  of  three  real  numbers  corresponds 
one  and  07ily  one  point. 

Locate  the  points :  (2,  3,  4),  (-  3,  2,  0),  (5,  0,-3),  (0,  0,  4), 
(0,-6,0),  (-5,  -8,  -2). 

290.  Distance  of  a  Point  from  the  Origin.  For  the  distance 
OP—r  (Fig.  121)  of  the  point  P{x,  y,  z)  from  the  origin  0  we 
have,  since  OP  is  the  diagonal  of  a  rectangular  parallelepiped 
with  edges  OQ'  =x,  OQ"  =y,  OQ"'  =  z: 


M 


^ 


t 


r 


r  =  -\Qi9'  +  2/^  +  z^. 

291.  Distance  between  two  Points.  The  distance  between 
the  two  points  Pj  (aJi,2/i>^i)  ^^^  A 
(^2 )  2/2  J  ^2)  can  be  found  if  the  coordi- 
nates of  the  two  points  are  given. 
For  (Fig.  123),  the  planes  through  P^ 
and  those  through  P^  parallel  to  the 
coordinate  planes  bound  a  rectangular 
parallelepiped  with  P^Pi  =  d  as  di- 
agonal ;  and  as  its  edges  are 

PiQ  =  x^-x,,   PiR=y2-yi 
we  find 

d  =  V(^2  -  ^if  +  (2/2  -  ViY  +  (^2  -  ^if- 

292.  Oblique  Axes.  The  position  of  a  point  P  in  space  can  also 
be  determined  with  respect  to  three  axes  not  at  right  angles.  The  coor- 
dinates of  P  are  the  segments  cut  off  on  the  axes  by  planes  through  P 


Fig.  122 


PS  =  z. 


XIII,  §  292]  COORDINATES  279 

parallel  to  the  coordinate  planes.    In  what  follows,  the  axes  are  always 
assumed  to  be  at  right  angles  unless  the  contrary  is  definitely  stated. 

EXERCISES 

1.  What  are  the  coordinates  of  the  origin  ?  What  can  you  say  of  the 
coordinates  of  a  point  on  the  axis  Ox  ?  on  the  axis  Oij  ?   on  the  axis  Oz  ? 

2.  What  can  you  say  of  the  coordinates  of  a  point  that  lies  in  the 
plane  Oxy  ?  in  the  plane  Oyz  ?  in  the  plane  Ozx  ? 

3.  Where  is  a  point  situated  when  a;  =  0  ?  when  0  =  0?  when 
x  =  y  =  0?  when  y  =  z'}  when  x  =  2?  when  0  =  —  3  ?  when  x  =  1 , 
2/  =  2? 

4.  A  rectangular  parallelepiped  lies  in  the  first  octant  with  three  of 
its  faces  in  the  coordinate  planes,  its  edges  are  of  length  a,  &,  c,  respec- 
tively ;  what  are  the  coordinates  of  the  vertices  ? 

5.  Show  that  the  points  (4,3,  5),  (2,  -1,3),  (0,1,7)  are  the 
vertices  of  an  equilateral  triangle. 

6.  Show  that  the  points  (-  1,  1,  3),  (—  2,  —  1,  4),  (0,  0,  5)  lie  on  a 
sphere  whose  center  is  (2,  —  3,  1).     What  is  the  radius  of  this  sphere  ? 

7.  Show  that  the  points  (6,  2,  -  5),  (2,  -  4,  7),  (4,  -  1,  1)  lie  on  a 
straight  line. 

8.  Show  that  the  triangle  whose  vertices  are  (a,  6,  c) ,  (6,  c,  a) ,  (c,  a,  6) 
is  equilateral. 

9.  What  are  the  coordinates  of  the  projections  of  the  point  (6,  3,  —  8) 
on  the  axes  of  coordinates  ?  What  are  the  distances  of  this  point  from  the 
coordinate  axes  ? 

10.  What  is  the  length  of  the  segment  of  a  line  whose  projections  on 
the  coordinate  axes  are  5,  3,  and  2  ? 

11.  What  are  the  coordinates  of  the  points  which  are  symmetric  to 
the  point  (a,  6,  c)  with  respect  to  the  coordinate  planes  ?  with  respect  to 
the  axes  ?  with  respect  to  the  origin  ? 

12.  Show  that  the  sum  of  the  squares  of  the  four  diagonals  of  a  rec- 
tangular parallelepiped  is  equal  to  the  sum  of  the  squares  of  its  edges. 


280  SOLID  ANALYTIC  GEOMETRY      [XIII,  §  293 

293.  Projection.  The  projection  of  a  point  on  a  plane  or 
line  is  the  foot  of  the  perpendicular  let  fall  from  the  point  on 
the  plane  or  line.  The  projection  of  a  rectilinear  segment  AB 
on  a  plane  or  line  is  the  intercept  A'B'  between  the  feet  of  the 
perpendiculars  AA',  BB'  let  fall  from  A,  B  on  the  plane  or 
line.  If  a  is  one  of  the  two  angles  made  by  the  segment  with 
the  plane  or  line  we  have 

A'B'  =  AB  cos  a. 

In  analytic  geometry  we  have  generally  to  project  a  vector, 
i.e.  a  segment  with  a  definite  sense,  on  an  axis,  i.e.  on  a  line 
with  a  definite  sense  (compare  §  19).  The  angle  a  is  then 
understood  to  be  the  angle  between  the  positive  senses  of 
vector  and  axis  (both  being  drawn  from  a  common  origin). 
The  above  formula  then  gives  the  projection  with  its  proper 
sign. 

Thus,  the  segment  OP  (Fig.  121)  from  the  origin  to  any 
point  P(x,  y,  z)  can  be  regarded  as  a  vector  OP.  Its  projec- 
tions on  the  axes  of  coordinates  are 
the  coordinates  x,  y,  z  of  P.  These 
projections  are  also  called  the  rec- 
tangular components  of  the  vector  OP, 
and  OP  is  called  the  resultant  of  the 
components  OQ',  OQ",  OQ'",  or  also 
of  OQ',  qP'",  P"'P. 

Similarly,  in  Fig.  123,  if  P^P^  be  Fig.  123 

regarded  as  a  vector,  the  projections  of  this  vector  P^P^  on  the 
axes  of  coordinates  are  the  coordinate  differences  x^  —  x^, 
2/2  —  2/i )  2^2  —  2i .     See  §  298. 

294.  Resultant.  The  proposition  of  §  19  that  tlie  sum  of 
the  projections  of  the  sides  of  an  open  polygon  on  any  axis  is 


<?" 


r 


XIII,  §  295] 


COORDINATES 


281 


equal  to  the  projectioyi  of  the  dosing  side  on  the  same  axis  and 
that  of  §  20  that  the  projection  of  the  resultant  is  equal  to  the 
sum  of  the  projections  of  its  components  are  readily  seen  to  hold 
in  three  dimensions  as  well  as  in  the  plane.  Analytically 
these  propositions  follow  by  considering  that  whatever  the 
points  P,{x^,  2/1,  z^),  P,(X2,  y^,  z^),  •••  P„K ,  y^,  z^)  in  space, 
the  sum  of  the  projections  of  the  vectors  PyPi,  P-iP^,  •••  Pn-i^n 
on  the  axis  Ox  is  : 

(x^-x,)-{-{x,-x,)-i-  •..  -\-(x^-x^_i)  =  x^-Xij 

where  the  right-hand  member  is  the  projection  of  the  closing 
side  or  resultant  PiP„  on  Ox.  Any  line  can  of  course  be  taken 
as  axis  Ox. 


295.  Division  Ratio.  Two  points  P-i{x^,  y^  z{)  and 
Pi  (^2  J  2/2 )  ^2)  being  given  by  their 
coordinates,  the  coordinates  x,  y,  z 
of  any  point  P  of  the  line  PyP^ 
can  he  found  if  the  division  ratio 
P^P/P^p2  =  k  is  known  in  ivhich 
the  point  P  divides  the  segment 
P,P,  (Fig  124). 

Let  Qi,  Q,  Qabe  the  projections 
of  Pj,  P,  P2  on  the  axis   Ox-,  as 
Q  divides  Q1Q2  in  the  same  ratio  k  in  which  P  divides  P1P2, 
we  have  as  in  §  3  : 

X  =^  X-^  ~\~  K  (3/2  —  "^1/* 

Similarly  we  find  by  projecting  on  Oy,  Oz : 

2/  =  2/1  +  A:  (2/2  -  yO,  z  =  Zi  +  k(z,  -  z^). 

If  k  is  positive,  P  lies  on  the  same  side  of  P^  as  does  Pg ;  if 
k  is  negative,  P  lies  on  the  opposite  side  of  Pj  (§  3). 


Fig.  124 


282  SOLID  ANALYTIC  GEOMETRY      [XIII,  §  296 

296.  Direction  Cosines.  Instead  of  using  the  cartesian 
coordinates  x,  y,  z  to  locate  a  point  P  (Fig.  125)  we  can  also 
use  its  radius  vector  r  =  OP,  i.e.  the  length  of  the  vector  drawn 
from  the  origin  to  the  point,  and  its  direction  cosines,  i.e.  the 
cosines  of  the  angles  a,  /3,  y,  made 
by  the  vector  OP  with  the  axes  Ox, 
Oy,  Oz.     We  haye  evidently 

ic  =  f  cos  a,  2/  =  r  cos  p,  z  =  r  cos  7. 

As  a  line  has  two  opposite  senses 
we    can    take    as    direction     cosines     jY         V 
of  any   line   parallel    to    OP    either  Fig.  125 

cos  a,  cos  p,  cos  y,  or  —  cos  a,  —  cos  ft,  —  cos  y. 

The  direction  cosines  cos  a,  cos  ft,  cos  y  of  a  vector  OP  are 
often  denoted  briefly  by  the  letters  Z,  m,  n,  respectively,  so 
that  the  coordinates  of  P  are 

x—lr,  y  =  mr,  z  =  nr. 

The  direction  cosines  of  any  parallel  line  are  then  I,  m,  n 
or  —I,  —m,  —  n. 

297.  Pythagorean  Relation.  The  sum  of  the  squares  of  the 
direction  cosines  of  any  line  is  equal  to  one. 

For,  the  equations  of  §  347  give  upon  squaring  and  adding 
since  7?  -\- y"^  -\- z^  =  r"^ : 

cos'^  ct  +  cos^  P  +  cos^  7  =  1? 
or 

Z2  +  m2  4-  ^i'  =  1 ; 

and  this  still  holds  when  I,  m,  n  are  replaced  by  —l,—m,—  n. 
Since  this  result  is  derived  directly  from  the  Pythagorean 
Theorem  of  geometry,  it  may  be  called  the  Pythagorean  Rela- 
tion between  the  direction  cosines.  Notice  that  I,  m,  n  can  be 
regarded  as  the  coordinates  of  the  extremity  of  a  vector  of 
unit  length  drawn  from  the  origin  parallel  to  the  line. 


XIII,  §297]  COORDINATES  283 

EXERCISES 

1.  Find  the  length  of  the  radius  vector  and  its  direction  cosines  for 
each  of  tlie  following  points  :  (5,  -  3,  2);  (-  3,  -  2,  1);  (-  4,  0,  8). 

2.  The  direction  cosines  of  a  line  are  proportional  to  1,  2,  3;  find 
their  values. 

3.  A  straight  line  makes  an  angle  of  30*^  with  the  axis  Ox  and  an 
angle  of  60°  with  the  axis  Oy  ;  what  is  the  third  direction  angle  ? 

4.  What  is  the  direction  of  a  line  when  Z  =  0  ?  when  Z  =  w  =  0  ? 

5.  What  are  the  direction  cosines  of  that  line  whose  direction  angles 
are  equal  ? 

6.  What  are  the  direction  cosines  of  the  line  bisecting  the  angle 
between  two  intersecting  lines  whose  direction  cosines  are  Z,  w,  n  and  I  , 
to',  w',  respectively  ? 

7.  Find  the  direction  cosines  of  the  line  which  bisects  the  angle 
between  the  radii  vectores  of  the  points  (3,  —  4,  2)  and  (—  1,  2,  3). 

8.  Three  vertices  of  a  parallelogram  are  (4,  3,  —2),  (7,  —  1,  4), 
(—2,  1,  —  4);  find  the  coordinates  of  the  fourth  vertex  (three  solutions). 

9.  In  what  ratio  is  the  line  drawn  from  the  point  (2,  —  5,  8)  to  the 
point  (4,  6,-2)  divided  by  the  plane  Ozx  ?  by  the  plane  Oxy  ?  At  what 
points  does  this  line  pierce  these  coordinate  planes  ? 

10.  In  what  ratio  is  the  line  drawn  from  the  point  (0,  5,  0)  to  the 
point  (8,  0,  0)  divided  by  the  line  in  the  plane  Oxy  which  bisects  the 
angle  between  the  axes  ? 

11.  Find  the  coordinates  of  the  midpoint  of  the  line  joining  the  points 
(4,  —  3,  8)  and  (6,  5,  —  9) .   Find  the  points  which  trisect  the  same  segment. 

12.  If  we  add  to  the  segment  joining  the  points  (4,  1,  2)  and  (—2, 
5,  7)  a  segment  of  twice  its  length  in  each  direction,  what  are  the  coordi- 
nates of  the  end  points  ? 

13.  Find  the  coordinates  of  the  intersection  of  the  medians  of  the  tri- 
angle whose  vertices  are  Pi  (xi ,  yi ,  Zx),  Ti  {xi^  yi,  zt),  Tz  (xs ,  yz  ,  zz). 

14.  Show  that  the  lines  joining  the  midpoints  of  the  opposite  edges 
of  a  tetrahedron  intersect  and  are  bisected  by  their  common  point. 

16.  Show  that  the  projection  of  the  radius  vector  of  the  point 
P(x, «/,  z)  on  a  line  whose  direction  cosines  are  l\  to',  w'  is  Vx  -f-  m^y  +  n^z. 


284 


SOLID  ANALYTIC  GEOMETRY      [XIII,  §  298 


r 


"298.   Projections.     Components  of  a  Vector.     If  two  points 

-PiC^'ij  Viy  ^\)  and  P2('^"2)  ?/2j  ^2)  are  given  by  their  coordinates, 

the  projections  of  the  vector,  P1P2  on 

the   axes,  or   what   amounts   to  the 

same,  on  parallels  to  the  axes  drawn 

through  Pj  (Eig.  126),  are  evidently 

(§  293)  :         • 

PyQ  =  X2-x,,    P,R  =  ^2  -  2/1, 
P^S  =  Z2  —  Zi. 

These  projections,  or  also  the  vectors 

PiQ,  QN,NP2y  are  called  the  rectangular  components  of  the 

vector  P1-P2  >  or  its  components  along  the  axes. 

If  d  is  the  length  of  the  segment  PiPo ,  its  direction  cosines  Z, 
m,  n  are  since  P^Q  is  perpendicular  to  P^Q,  P2R  to  P^E,  P2S 
to  P,S: 


Fig.  126 


1  = 


3/2 


?/o  —  V, 


These  relations  can  also  be  written  in  the  form : 

X2—Xy  ^  ^2  —   .Vl  ^  ^2  —  '^!  ^  fl 

I  m  n 


(li,m2,nt) 


299.  Angle  between  two  Lines.  Iftlie  directions  of  two  lines 
are  given  by  their  direction  cosines  li ,  mi ,  n^  and  I2 ,  ^2 ,  ng ,  the 
angle  ij/  between  the  two  lines  is  given 
by  the  formula 

cos  x|/  =  I1I2  +  mitn^2  -I-  nin2- 

For,  drawing  through  the  origin 
two  lines  of  direction  cosines  li ,  mi , 
ni  and  ^2  >  ^2  >  ^2  and  taking  on  the    jj/^        ^' 
former  a  vector  OPi  of  unit  length,  Fig-  127 

the  projection  OP  of  OPi  on  the  other  line  is  equal  to  the 


r 

/ 

/  ^v^^     ^^l^miMii 

0 

£^>-'''^V             y 

5^ 

\Jy 

XIII,  §  3011  COORDINATES  285 

cosine  of  the  required  angle  ij/.  On  the  other  hand,  OPi  has 
h)  *^i)  %  ^s  components  along  the  axes ;  hence,  by  §  294 : 
cos  {{/  =  I1I2  +  mimg  +  711112. 
Two  intersecting  lines  (or  any  two  parallels  to  them)  make 
two  angles,  say  xf/  and  ir  —  \p.  But  if  the  direction  cosines  of 
each  line  are  given,  a  definite  sense  has  been  assigned  to  each 
line,  and  the  angle  between  the  lines  is  understood  to  be  the 
angle  between  these  senses. 

300.  Conditions  for  Parallelism  and  for  Perpendicularity. 

If,  in  particular,  the  lines  are  parallel,  we  have  either  l^  ==  I2, 
mi  =  m2 ,  Til  =  712,  or  li  =  — 12,  771^  =  —  m2,  tii  =  —  Wj ;  hence  in 
either  case  l,^ni,^7i. 

This  then  is  the  condition  of  parallelism  of  two  lines  whose 
direction  cosines  are  ^1,  m^,  n^  and  I2,  m,,  712. 

If    the    lines    are  perpendicular,    i.e.    if  j/^=i7r,   we   have 
cos  1/^  =  0;  hence  the  condition  of  perpendicularity  of  two  lines 
whose  direction  cosines  are  li,  mi,  n^  and  I2 ,  m2,  712  is 
I1I2  +  mim2  +  ni7i2  =  0. 

301.  The  formula  of  §  299  gives 

sm2  xp  =  1  -  cos2  xf/  =  1  —  (I1I2  4-mim2  +  ^1^2)^. 
As  (§  297)  (Zi2  +  mi2  +  ni^)(l2^  +  m2^  +  n2'^)=  1,  we  can  write  this  ex- 
pression in  the  form 

sin2  xl/  =   ^^^  "^  *^^^  "^  ^^^  ^^^2  '^  ''^^'"^^  "*■  '*^^2 

hh  4-  mim2  +  nin2      l-^  +  m-^  +  ^2^ 

which,  by  Ex.  3,  p.  45,  can  also  be  expressed  as  follows  : 

h    WI2I 

The  direction  (I,  m,  w)  perpendicular  to  two  given  different  directions 
(l\ ,  wi ,  ni)  and  (^2 ,  W2 ,  W2)  is  found  by  solving  the  equations  (§  300) 
III  +  m\m  +  Jiiw  =  0, 
hi  +  wi2W  +  n2n  =  0, 


sin^i// 


mi     ?ii 

•i 

n\    h 

+ 

+ 

m2    ^2 

W2      ^2 

286 

whence 


SOLID  ANALYTIC  GEOMETRY      [XIII,  §  301 


{ 

m 

n 

mi 

Wl 

7ll      h 

h    mi 

W2 

W2 

W2      h 

h      W2 

If  we  denote  by  k  the  common  value  of  these  ratios,  we  have 

h     mi 

h      Wi2 


1  = 


mi    wi 

W2       W2 


Wl      ii 
W2      Z2 

substituting  these  values  in  the  relation  (§  297)  P-  +  m^  +  ^2  _  1  and, 
observing  the  preceding  value  of  sin  ^,  we  find : 


l=± 


mi     ni 
m^    712 


m  =± 


li     mi 

h      Wl2 


sin  ^  sin  \p 

where  \p  is  the  angle  between  the  given  directions. 


sin^ 


302.   Three  directions  (Zi,  wii,  Wi),  (Z2,  wi2,  ih),  (Izi  WI3,  W3)  are  com- 
planar,  i.e.  parallel  to  the  same  plane,  if  there  exists  a  direction  (Z,  m,  w) 
perpendicular  to  all  three.    This  will  be  the  case  if  the  equations 
hi  +  wiim  +  Tiin  =  0, 
hi  +  W2m  +  n2n  —  0, 
?3?  +  wi3m  +[?i3n  =  0 
have  solutions  not  all  zero  ;  hence  the  condition  of  complanarity 
h    mi     ni 
h    mi    112. 
h    mz    Wa 

EXERCISES 

1.  Find  the  length  and  direction  cosines  of  the  vector  drawn  from  the 
point  (5,  —2,  1)  to  the  point  (4,  8,  —  6)  ;  from  the  point  (a,  &,  c)  to  the 
point  (  —  a,  —6,  — c)  ;  from  (  —  a,  —h,  — c)  to  (a,  ?),  c). 

2.  Show  that  when  two  lines  with  direction  cosines  ?,  w,  7i  and 
l\  w',  7i',  respectively,  are  parallel,  IV  +  wi>^'  +  wn'  =±1.        2 

3.  Show  that  when  two  lines  with  direction  cosines  proportional  to 
a,  6,  c,  and  a',  6',  c',  are  perpendicular  aa'  +  &&'+  cc'  =  0  ;  and  when  the 
lines  are  parallel  a/ a'  =h/h'  =  c/c'. 

4.  Show  that  the  points  (5,  2,  -3),  (6,  1,  4),  (-2,  -3,  6), 
(—1,  —  4,  13)  are  the  vertices  of  a  parallelogram. 


XIII,  §  303]  COORDINATES  287 

5.  Show  by  direction  cosines  that  the  points  (6,  —3,  5),  (8,  2,  2), 
(4,  —8,  8)  lie  in  a  line. 

6.  Find  the  angle  between  the  vectors  from  (5,  8,  —  2)  to  (—2,  6,-1) 
and  from  (8,  3,  5)  to  (1,  1,  -6). 

7.  Find  the  angles  of  the  triangle  whose  vertices  are  (5,  2,  1), 
(0,3,  -1),(2,  -1,7). 

8.  Find  the  direction  cosines  of  a  line  which  is  perpendicular  to  two 
lines  whose  direction  cosines  are  proportional  to  2,  —3,  4,  and  5,  2,  —1, 
respectively. 

9.  Derive  the  formula  of  §  299  by  taking  on  each  line  a  vector  of  unit 
length,  OPi  and  OP2,  and  expressing  the  distance  P1P2  first  by  the 
cosine  law  of  trigonometry,  then  by  §  292,  and  equating  these  expressions. 

10.  Find  the  rectangular  components  of  a  force  of  12  lb.  acting  along 
a  line  inclined  at  60°  to  Ox  and  at  45°  to  Oy. 

11.  Find  the  resultant  of  the  forces  OPi,  OP2,  OP3,  OP4  if  the  co- 
ordinates of  Pi,  P2,  P3,  P4,  with  O  as  origin,  are  (3,  —1,  2),  (2,  2,-1), 
(-1,2,1),  (-2,  3,  -4). 

12.  If  any  number  of  vectors,  applied  at  the  origin,  are  given  by  the 
coordinates  x,  y,  z  of  their  extremities,  the  length  of  the  resultant  H  is 
\/(Sx)2  +  {^yy^  +  (Ss)"-^  (see  Ex.  9,  p.  21),  and  its  direction  cosines 
are  S  xjB,  S  yjB,  S  zIB. 

13.  A  particle  at  one  vertex  of  a  cube  is  acted  upon  by  seven  forces 
represented  by  the  vectors  from  the  particle  to  the  other  seven  vertices  ; 
find  the  magnitude  (length)  and  direction  of  the  resultant. 

14.  If  four  forces  acting  on  a  particle  are  parallel  and  proportional  to 
the  sides  of  a  quadrilateral,  the  forces  are  in  equilibrium,  i.e.  their  resultant 
is  zero.     Similarly  for  any  closed  polygon. 

303.  Translation  of  Coordinate  Trihedral.  Let  x,  y,  z  be 
the  coordinates  of  any  point  P  with  respect  to  the  trihedral 
formed  by  the  axes  Ox,  Oy,  Oz  (Fig.  128).  If  parallel  axes 
^i^ij  ^lVu  ^1%  t)6  drawn  through  any  point  Oi(a,  6,  c),  and  if 
^j>  2/ij  ^1  ^^^  the  coordinates  of  P  with  respect  to  the  new  tri- 


288 


SOLID  ANALYTIC  GEOMETRY      [XIII,  §  303 


hedral  OxX{y^Zi,  then  the  relations  between  the  old  coordinates 
X,  y,  z,  and  the  new  coordinates  Xy,  y^,  z^  of  one  and  the  same 
point  P  are  evidently 

x  =  a-}-x^,        y  =  b  +  7ji,       z  =  c  +  z^. 

The  coordinate  trihedral  has  thus 
been  given  a  translation,  represented 
by  the  vector  00^^.  This  operation 
is  also  called  a  transformation  to 
parallel  axes  through  Oi-  Fig.  128 

.  304.  Area  of  a  Triangle.  Any  two  vectors  OPi,  OP2  drawn  from 
the  origin  determine  a  triangle  OP1P2,  whose  area  A  can  easily  be  ex- 
pressed if  the  lengths  ri ,  r2  and  direction  cosines 
of  the  vectors  are  given.  For,  denoting  the  angle 
Pi  OP2  by  \p  we  have  for  the  area  A  r 

A  =  \  riTi  sin  ^, 

where  sin  ^  can  be  expressed  in  terms  of  the  direc- 
tion cosines  by  §  301. 


yQz 


Fig.  129 


305.  Moment  of  a  Force.  Such  areas  are  used  in  mechanics  to 
represent  the  moments  of  forces.  The  moment  of  a  force  about  a  point  O 
is  defined  as  the  product  of  the  force  into  the 
perpendicular  distance  of  0  from  the  line  of 
action  of  the  force.  Thus,  if  the  vector  P1P2 
(Fig.  130)  represent  a  force  (in  magnitude, 
direction,  and  sense)  the 'moment  of  this  force 
about  the  origin  0  is  equal  to  twice  the  area 
of  the  triangle  OP1P2,  i.e.  to  the  area  of  the 
parallelogram  OP1P2P3,  where  OP3  is  a  vector 
equal  to  the  vector  P1P2.  Fig.  130 

It  is  often  more  convenient  to  represent  this  moment  not  by  such  an 
area,  but  by  a  vector  OQ,  drawn  from  O  at  right  angles  to  the  triangle, 
and  of  a  length  equal  to  the  number  that  represents  the  moment.  If  the 
body  on  which  the  force  acts  could  turn  freely  about  this  perpendicular 
the  moment  would  represent  the  turning  effect  of  the  force  P1P2. 


XIII,  §  306] 


COORDINATES 


289 


The  'sense  of  this  vector  that  represents  the  inoiuent  is  taken  so  as  to 
make  the  vector  point  toward  that  side  of  the  plane  of  the  triangle  from 
which  the  force  P1P2  is  seen  to  turn  counterclockwise. 

306.  If  we  square  the  expression  found  in  §  304  for  the  area  of  the 
triangle  OP1P2  and  substitute  for  sin'^^j/  its  value  from  §  301,  we  find  : 


A^ 


=  \n^r2^(^ 


mi 

Wi 

2 

ni 

h 

2       h    mi 

m2 

nz 

+ 

Wo 

h 

■"    I2    m2 

1 


Hence  A^  is  the  sum  of  the  squares  of  the  three  quantities 


Ax  =  i  riVi 


mi     Wi 
1712     W2 


A, 


ni     h 
n.2     h 


A^=\  rira 


which  have  a  simple  geometrical  and  mechanical  interpretation.     For,  as 
the  coordinates  of  Pi ,  Po  are 

xi  =  hn,      yi  =  wiiri,     zi  =  niVi, 
Xi  =  hr2',      y2  =  m2r2,     Z2  =  ^2^2, 


we  have, 

e.g., 

A.=  l 

hn    rrnn 

hrz    wi2r2 

=  i 

xi    yi 
X2    2/2 

and  as  Xi ,  yi  and  X2 ,  2/2  are  the  coordinates  of  the  projections  ^1 ,  ^2  of 
Pi ,  P2  on  the  plane  Oxy,  Az  represents  (§  12)  the  area  of  the  triangle 
0Q\Q2-,  i-e.  the  projection  on  the  plane  Oxy  of  the  area  OP1P2.  Sim- 
ilarly, Aj.  and  Ay  are  the  projections  of  the  area  OP1P2  on  the  planes 
Oyz  and  Ozx,  respectively.  As  any  three  mutually  rectangular  planes 
can  be  taken  as  coordinate  trihedrals,  our  formula  A^  =  A^  +  A^  +  A^ 
means  that  the  square  of  the  area  of  any  triangle  is  equal  to  the  sum  of 
the  squares  of  its  projections  on  any  three  mutually  rectangular  planes. 

In  mechanics,  2  A^  is  the  moment  of  the  projection  Qi  Q2  of  the  force 
P1P2  about  0,  or  what  is  by  definition  the  same  thing,  the  moment  of 
P1P2  about  the  axis  Oz.  Similarly,  for  2^^,  2  Ay.  The  proposition 
means,  therefore,  that  the  moments  of  P1P2  about  the  axes  Ox,  Oy,  Oz 
laid  off  as  vectors  along  these  axes  can  be  regarded  as  the  rectangular 
components  of  the  moment  of  P1P2  about  the  point  0 ;  in  other  words, 
2  ^,,  2  Ay,  2  Ag  are  the  components  along  Ox,  Oy,  Oz  of  that  vector 
2  ^  (§  305)  which  represents  the  moment  of  P1P2  about  O. 
u 


290 


SOLID  ANALYTIC   GEOMETRY      [XIII,  §  307 


307.  Polar  Coordinates.  The  position  of  any  point  P  {¥\\ 
131)  can  also  be  assigned  by  its 
radius  vector  OP=r,  i.e.  the  dis- 
tance of  P  from  a  fixed  origin  or 
pole  O,  and  two  angles  :  the  colati- 
tude  6,  i.e.  the  angle  NOP  made 
by  OP  with  a  fixed  axis  ON,  the 
2)olar  axis,  and  the  longitude  (f>, 
i.e.  the  angle  AOP'  made  by  the 
plane  of  9  with  a  fixed  plane 
NOA  through  the  polar  axis,  the 
initial  meridian  plane. 

A  given  radius  vector  r  confines  the  point  P  to  the  sphere 
of  radius  r  about  the  pole  0.  The  angles  0  and  <^  serve  to 
determine  the  position  of  P  on  this  sphere.  This  is  done  as 
on  the  earth's  surface  except  that  instead  of  the  latitude,  which 
is  the  angle  made  by  the  radius  vector  with  the  plane  of  the 
equator  AP',  we  use  the  colatitude  or  polar  distance  0  =  NOP. 

The  quantities  r,  6,  and  <^  are  the  polar  or  spherical  coordi- 
nates of  P.  After  assuming  a  point  0  as  pole,  a  line  ON 
through  0,  with  a  definite  sense,  as  polar  axis,  and  a  (half-) 
plane  through  this  axis  as  initial  meridian  plane,  every  point 
P  has  a  definite  radius  vector  r  (varying  from  zero  to  infinity), 
colatitude  6  (varying  from  0  to  tt),  and  a  definite  longitude  <^ 
(varying  from  0  to  2  tt).  The  counterclockwise  sense  of  rotation 
about  the  polar  axis  is  taken  as  the  positive  sense  of  <^. 


308.  Transformation  from  Cartesian  to  Polar  Coordinates- 

The  relations  between  the  cartesian  coordinates  x,  y,  z  and  the 
polar  coordinates  r,  6,  <j>  of  any  point  P  appear  directly  from 
Fig.  132.  If  the  axis  Oz  coincides  with  the  polar  axis,  the 
plane  Oxy  with  the  equatorial  j^lane,  i.e.  the  plane  through  the 


XIII,  §  308]  COORDINATES  291 

pole  at  right  angles  to  the  polar  axis,  while  the  plane  Ozx  is 
taken  as  initial  meridian  plane,  the  pro-  ^ 

jections  of  OP  =  r  on  the  axis  Oz  and  ^ 

on  the  equatorial  plane  are 

OR  =  rGO^e,  OQ  =  r sine. 

Projecting  OQ  on  the  axes  Ox,  Oy,we 
find  Fig.  132 

x=r  sin  6  cos  <^,     y  =  r  sin  6  sin  <^,     z  =  r  cos  6. 

Also     r  =  Vx^-\-y^  -\-  Z-,  cosO  =  —       ^  tan<^  =  '^. 

Va^  -h  y2  ^  ;32  X 

EXERCISES 

1.  Find  the  area  of  the  triangle  whose  vertices  are  (a,  0,  0),  (0,  6,  0), 
(0,  0,  c). 

2.  Find  the  area  of  the  triangle  whose  vertices  are  the  origin  and  the 
points  (3,  4,  7),  (-  1,  2,  4). 

3.  Find  the  area  of  the  triangle  whose  vertices  are  (4,  —  3,  2), 
(6,4,4),  (-5,  -2,  8). 

4.  The  cartesian  coordinates  of  a  point  are  1,  VS,  2\/3  ;  what  are  its 
polar  coordinates  ? 

5.  If  r  =  5,  0  =  i  TT,  0  =  ^  TT,  what  are  the  cartesian  coordinates  ? 

6.  The  earth  being  taken  as  a  sphere  of  radius  3962  miles,  what  are 
the  polar  and  cartesian  coordinates  of  a  point  on  the  surface  in  lat.  42°  17' 
N.  and  long.  83°  44'  W.  of  Greenwich,  the  north  polar  axis  being  the  axis 
Oz  and  the  initial  meridian  passing  through  Greenwich  ?  What  is  the 
distance  of  this  point  from  the  earth's  axis  ? 

7.  Find  the  area  of  the  triangle  whose  vertices  are  (0, 0, 0) ,  (ri,  0i,  0i), 
(ra,  02,  02). 

8.  Express  the  distance  between  any  two  points  in  polar  coordinates. 

9.  Find  the  area  of  any  triangle  when  the  cartesian  coordinates  of  the 
vertices  are  given. 

10.  Find  the  rectangular  components  of  the  moment  about  the  origin 
of  the  vector  drawn  from  (1,  —  2,  3)  to  (3,  1,  —  1). 


CHAPTER   XIV 

THE  PLANE   AND   THE   STRAIGHT  LINE 

PART   I.     THE   PLANE 

309.  Locus  of  One  Equation.  In  plane  analytic  geometry 
any  equation  between  the  coordinates  a:,  y  or  r,  <^  of  a  point  in 
general  represents  a  plane  curve.  In  particular,  an  equation  of 
the  first  degree  in  x  and  y  represents  a  straight  line  (§  30); 
an  equation  of  the  second  degree  in  x  and  y  in  general  repre- 
sents a  conic  section  (§  245). 

In  solid  analytic  geometry  any  equation  between  the  coordi- 
nates ic,  y,  z  or  ?',  d,  <^  of  a  point  in  general  represents  a  surface. 
Thus,  if  any  equation  in  x,  y,  z, 

F{x,y,z)  =  0, 

be  imagined  solved  for  z  so  as  to  take  the  form 

2;=/(a;,  y), 

we  can  find  from  this  equation  to  every  point  (a;,  y)  in  the 
plane  Oa^j  one  or  more  ordinates  z  (which  may  of  course  be 
real  or  imaginary),  and  the  locus  formed  by  the  extremities  of 
the  real  ordinates  will  in  general  form  a  surface.  It  may  how- 
ever happen  in  particular  cases  that  the  locus  of  the  equation 
F(x,  y,  z)  =  0,  i.e.  the  totality  of  all  those  points  whose  coordi- 
nates x,  yj  z  when  substituted  in  the  equation  satisfy  it,  con- 
sists only  of  isolated  points,  or  forms  a  curve,  or  that  there  are 
no  real  points  satisfying  the  equation. 

Similar    considerations    apply    to    an    equation    in    polar 

coordinates 

F(r,  $,<!>)  =0. 
202 


XIV,  §311]  THE  PLANE  •      293 

310.  Locus  of  Two  Simultaneous  Equations.  Two  simulta- 
neous equations  in  x,  y,  z  (or  in  the  polar  coordinates  r,  6,  </>) 
will  in  general  represent  a  curve  in  space,  namely,  the  inter- 
section of  the  two  surfaces  represented  by  the  two  equations 
separately. 

Thus,  in  the  present  chapter,  we  shall  see  that  an  equation  of 
the  first  degree  in  x,  y,  z  represents  a  plane  and  that  therefore 
two  such  equations  represent  a  straight  line,  the  intersection  of 
the  two  planes.  In  chapters  XV  and  XVI  we  shall  discuss 
loci  represented  by  equations  of  the  second  degree,  which  are 
called  quadric  surfaces. 

311.  Equation  of  a  Plane.  Every  equation  of  the  first  degree 
in  X,  y,  z  represents  a  plane.  The  plane  is  defined  as  a  surface 
such  that  the  line  joining  any  two  of  its  points  lies  completely 
in  the  surface.  We  have  therefore  to  show  that  if  the  general 
equation  of  the  first  degree 

(1)  Ax-{-By-\-Cz+D  =  0 

is  satisfied  by  the  coordinates  of  any  two  points  Pi(x^y  y^  z^ 

and  P2fe>  Vi)  %)>  ^•^'  if 

^  +  ^yi  +  Czi  +  Z>  =  0, 

Ax^-\-By^+Cz^-\-D=^0, 

then    (1)    is    satisfied    by   the    coordinates    of    every    point 
P{x,  y,  z)  of  the  line  PiP^. 

Now,  by  §  295,  the  coordinates  of  every  point  of  the  line 
P^Pi  can  be  expressed  in  the  form 

x  =  x^-\-k(x^-x;),  y  =  yi  +  k(y2~yi),  z  =  Zj  +  A:(% - Zj), 
where  k  is  the  ratio  in  which  P  divides  PiP^j  i-e. 

k  =  P,P/P,P,. 
We  have  therefore  to  show  that 
A[x^  +  kix^  -  a^)]  +  B[y^  +k{y,  -yO]  +  C[_z,-\-k{z,-z,)-]  +/)=0, 


(2) 


294  SOLID  ANALYTIC  GEOMETRY       [XIV,  §  311 

■whatever  the  value  of  k.  Adding  and  subtracting  kD,  we  can 
write  this  equation  in  the  form 

(1  -  k)(Ax,-{-  By^-^  Cz:,  +  D)  -i-kiAx^-^  By,-{-  Cz2-\-  D)  :=0', 

and  this  is  evidently  true  for  any  k,  owing  to  the  conditions  (2). 

312.  Essential  Constants.  The  equation  (1)  will  still  rep- 
resent the  same  plane  when  multiplied  by  any  constant  differ- 
ent from  zero.  Since  A,  h,  C  cannot  all  three  be  zero,  we 
can  divide  (1)  by  one  of  these  constants ;  it  will  then  contain 
not  more  than  three  arbitrary  constants.  We  sa}^  therefore 
that  the  general  equation  of  a  plane  contains  th7'ee  essential 
C07istants.  This  corresponds  to  the  geometrical  fact  that  a 
plane  can,  in  a  variety  of  ways,  be  determined  by  three  condi- 
tions, such  as  the  conditions  of  passing  through  three  points, 
etc. 

313.  Special  Cases.  If,  in  equation  (1),  D  =  0,  the  plane 
evidently  passes  through  the  origin. 

If,  in  equation  (1),  0=0,  so  that  the  equation  is  of   the 

form 

Ax  +  By-\-D  =  0, 

this  equation  represents  the  plane  perpendicular  to  the  plane 
Oxy  and  passing  through  the  line  whose  equation  in  the 
plane  Oayy  is  xlx  -i-By  -\-  D  =  0.  For,  the  equation  Ax  +  By 
-h  Z>  =  0  is  satisfied  by  the  coordinates  of  all  points  (x,  y,  z) 
whose  x  and  y  are  connected  by  the  relation  Ax -\- By -\- D  =  0 
and  whose  z  is  arbitrary,  but  it  is  not  satisfied  by  the  coordi- 
nates of  any  other  points.  Similarly,  if  ^  =  0  in  (1),  the  plane 
is  perpendicular  to  Ozx ;  if  ^  =  0,  the  plane  is  perpendicular  to 
Oyz. 

If  5  =  0  and  0=0  in  (1),  the  equation  obviously  represents 
a  plane  perpendicular  to  the  axis  Ox ;  and  similarly  when  0 
and  A,  or  A  and  B  are  zero. 


XIV,  §315]  THE   PLANE  295 

Notice  that  the  line  of  intersection  of  (1)  with  the  plane 
Oxy,  for  instance,  is  represented  by  the  simultaneous  equations 
Ax  +  By-\-Cz  +  D=^0,z  =  (i. 

314.  Intercept  Form,     li   D^O  the  equation   (1)  can  be 
divided  by  Z>;  it  then  assumes  the  form 

D         D^      D 

If  A,  B,  C  are  all  different  from  zero,  this  equation  can  be 

written 

^        I        y        I        ^       ^1 
-D/A^  -D/B^  -D/C       ' 

or,  putting  -  D/A  =  a,  -  D/B  =  b,  -  D/C=  c : 

(3)  ^  +  |  +  ?=1. 

a      b      c 

In  this  equation,  called  the  intercept  form  of  the  equation 

of  a  plane,  the  constants  a,  b,  c  are  the  intercepts  made  by  the 

plane  on  the  axes  Ox,  Oy^  Oz  respectively.     For,  putting,  for 

instance,  y  =  0  and  z  =  0,  we  find  x  =  a\  etc. 

315.  Plane  through  Three  Points.    If  the  plane 

Ax  +  By+Cz  +  D  =  0 
is  to  pass  through  the  three  points  Px{x^,  yi,  Zi),  ^2(^2?  2/2?  %)> 
Ps(xs,  2/3,  23),  the  three  conditions 

'  Ax,  +  By,-\-Cz,+D=0, 
Ax,^  +  %2  +  Cz.  4-  Z)  =  0, 
Ax,  +  By,-^Cz,  +  D  =  0 
must  be  satisfied.     Eliminating  A,  B,  (J,  D  between  the  four 
linear  homogeneous  equations  (compare  §  75)  we  find  the  equa- 
tion of  the  plane  passing  through  the  three  points  in  the  form 
X     y     z     1 
^1     2/1     ^i     1 
»2     2/2     2,     1 

•^3       2/3        '^Z        -I 


=  0. 


296 


SOLID  ANALYTIC  GEOMETRY       [XIV,  §  315 


EXERCISES 

1.  Find  the  intercepts  made  by  the  following  planes  : 

(a)  4  a;  +  12  ?/  +  3  ;2  =  12  :  (b)  15  x  -  6 1/  +  10  ^  +  30  =  0  ; 

(c)  x-y  -\-3-l=0;  (d)  x-^2y  -\-S.z  +  4  =  0, 

2.  Interpret  the  following  equations : 

(a)  x+y  +  z  =  l;  (b)  6y  -  ^  z  =  12  ; 

(c)  x  +  y=0;  (d)  6y  +  l2=0. 

'"  3.  Find  the  plane  determined  by  the  points  (2,  1,  3),  (1,  —5,0), 
(4,6,  -1). 

4.  Write  down  the  equation  of  the  plane  whose  intercepts  are  3,  2,  —  5. 

5.  Find    the    intercepts   of    the    plane    passing    through    the   points 
(3,  -1,4),  (6,2,-3),  (-1,  -2,  -3). 

6.  If  planes  are  parallel  to  and  a  distance  a  from  the  coordinate  planes, 
what  are  their  intercepts  ?     What  are  their  equations  ? 

7.  Show    that    the    four    points    (4,3,3),    (4,-3,-9),    (0,0,3), 
(2,  1,  2)  lie  in  a  plane  and  find  its  equation. 

316.   Normal   Form.     The  position  of  a  plane  in  space  is 
fully  determined  by  the  length  p  =  ON  (Fig.  133)  of  the  per- 
pendicular let  fall  from  the  origin 
on  the  plane  and  the  direction  co- 
sines I,  m,  n  of  this  perpendicular 
regarded  as  a  vector  ON.  Let  Pbe 
any  point  of  the  plane  and  OQ=x, 
QR  =  y,  HP—  z  its  coordinates ;  as 
the  projection  of  the  open  polygon 
OQRP  on    ON   is   equal   to    ON 
(§  294)  we  have 
(4)  Ix -\- my -\- nz  =  p. 

This  equation  is  called  the  normal  form  of  the  equation  of  a 
plane.  Observe  that  the  number  p  is  always  positive,  being 
the  distance  of  the  plane  from  the  origin,  or  the  length  of  the 
vector  ON     Hence  Ix  +  my  -\-  nz  is  always  positive. 


Fig.  133 


XIV,  §317]  THE  PLANE  297 

317.  Reduction  to  the  Normal  Form.  The  equation  Ax  + 
By  -{-  Cz  -\-  D  =  0  is  in  general  not  of  the  form  lx+my+nz=p 
since  in  the  latter  equation  the  coefficients  of  x,  y,  z,  being  the 
direction  cosines  of  a  vector,  have  the  property  that  the  sum 
of  their  squares  is  equal  to  1,  while  A^-\-B--{-  C^  is  in  general 
not  equal  to  1.  But  the  general  equation  can  be  reduced  to 
the  normal  form  by  multiplying  it  by  a  constant  factor  k 
properly  chosen.     The  equation 

kAx  +  kBy  +  kCz  +  kD  =  0 
evidently  represents   the   same  plane   as   does   the   equation 
Ax  +  By  H-  Cz  +  D  =  0;  and  we  can  select  k  so  that 

{kAy  +  (kBf  +  (kCy  =  1,     viz.  k=  ^ 


±VA'-\-B^-\-C^ 
As  in  the  normal  form  the  right-hand  member  p  is  positive 
(§  316)  the  sign  of  the  square  root  should  be  selected  so  that 
kD  becomes  negative. 

The  normal  fonn  is  therefore  obtained  by  dividing  the  equation 
Ax  +  By  -\-  Gz-^D  =  Oby  ±  VA^  +  B^ -^  G^  according  as D  is 
negative  or  positive. 

It  follows  at  the  same  time  that  the  direction  cosines  of  any 
normal  to  the  plane  Ax  -\-  By  +  Cz  -\-  D  =  0  are  proportional 
to  Ay  Bf  C,  viz. 

;  A  B 

1=  ,    m  = 


±V^2  +  52+    C2  ±  V^2_^52_^02 


±VA''-\-B'-\-  (? 
and  that  the  distance  of  the  plane  from  the  origin  is 


P 


±  VA?  -f-  52  ^  C2 
the  upper  sign  of  the  square  root  to  be  used  when  D  is  nega- 
tive, the  lower  when  D  is  positive. 


298 


SOLID  ANALYTIC  GEOMETRY       [XIV,  §  318 


Fig.  134 


318.   Distance  of  Point  from  Plane.     Let  Ix  -{-  my  -i-nz  =p 
be  the  equation  of  a  plane  in  the  normal  form,  Pi{xi,  y^,  z^ 
any  point  not  on  this  plane  (Fig.  134).     The  projection  OS  of 
the  vector  OP^  on  the  normal  to  the 
plane  being  equal  to  the  sum  of  the 
projections  of  its  components  0Q  = 
Xij  QR  =  2/i,  RP\  =  2;i,  we  have 
OS  =  lx^-\-  myi  +  nzi. 

Hence  the  distance  d  of  Pi  from  the 
plane,  which  is  equal  to  NS,  will  be 

d  =  OS  —  0N=  Ix^  +  myi  +  nzy^  —  p. 

If  this  expression  is  negative,  the  point  P^  lies  on  the  same 
side  of  the  plane  as  does  the  origin ;  if  it  is  positive,  the  point 
Pi  lies  on  the  opposite  side  of  the  plane.  Any  plane  thus  di- 
vides space  into  two  regions,  in  one  of  which  the  distance  of 
every  point  from  the  plane  is  positive,  while  in  the  other  the 
distance  is  negative.  If  the  plane  does  not  pass  through  the 
origin,  the  region  containing  the  origin  is  the  negative  region ; 
if  it  does,  either  side  can  be  taken  as  the  positive  side. 

To  find  the  distance  of  a  point  Pi(.'«i,  y^  Zi)  from  a  plane 

given  in  the  general  form 

Ax-\-By  +  Cz-{-D  =  0, 

we  have    only  to  reduce  the   equation   to  the  normal    form 

(§  317)  and  then  to  substitute  for  x,  y,  z  the  coordinates  Xi,  yi, 

Zi  of  Pi;  thus 

^  ^  ^Xi  +  By,  +  Cz^^-  D 

the  square  root  being  taken  with  +  or  —  according  as  D  is 
negative  or  positive. 

Notice  that  d  is  the  distance  from  the  plane  to  the  point 
Pi ,  not  from  Pi  to  the  plane. 


XIV,  §  320]  THE  PLANE  299 

319.  Angle  between  Two  Planes.  As  two  intersecting 
planes  make  two  angles  whose  sum  =  tt,  we  shall,  to  avoid  any 
ambiguity,  define  the  angle  between  the  planes  as  the  angle 
between  the  perpendiculars  (regarded  as  vectors)  drawn  from 
the  origin  to  the  two  planes. 

If  the  equations  of  the  planes  are  given  in  the  normal  form, 

kx  +  m^  +  n^z  =  pi, 
l^  +  m^  +  n.^  =  2)2, 
we  have,  by  §  299,  for  the  angle  if/  between  the  planes : 

cos  \p  =  IJ2  +  wi^ma  +  ri^ng. 
If  the  equations  of  the  planes  are  in  the  general  form, 

A^  +  B^  +  C2Z  +  A  =  0, 
we  find  by  reducing  to  the  normal  form  (§  317) : 


cos  l{/  = 


A,A,-\-B,B2+C,C, 


±  VA'  +  B,'  +  0^2 .  ±  VA^  +  ^^2  +  c^2 


320.  Bisecting  Planes.  To  find  the  equations  of  the  two 
planes  that  bisect  the  angles  formed  by  two  intersecting  planes 
given  in  the  normal  form, 

liX  +  Miy  +  n^z  —pi  =  0,  l^-\-  m<^  +  n2Z  —  pg  =  0, 

observe  that  for  any  point  in  either  bisecting  plane  its  distances 
from  the  two  given  planes  must  be  equal  in  absolute  value. 
Hence  the  equations  of  the  required  planes  are 

l^x  +  m^y  -\-n^z—p^  =  ±  Q^  +  m^  +  n^z  —  ^2)- 
To  distinguish  the  two  planes,  observe  that  for  the  plane  that 
bisects  that  pair  of  vertical  angles  which  contains  the  origin 
the  perpendicular  distances  are  in  the  one  angle  both  positive, 
in  the  other  both  negative;  hence  the  plus  sign  gives  this 
bisecting  plane. 


300  SOLID  ANALYTIC  GEOMETRY       [XIV,  §  320 

If  the  equations  of  the  planes  are  given  in  the  general  form, 

first  reduce  the  equations  to  the  normal  form  (§  317). 

EXERCISES 

1.  A  line  is  drawn  from  the  origin  perpendicular  to  the  plane 
a;  —  ?/  —  50  —  10  =  0;   what  are  the  direction  cosines  of  this  line  ? 

2.  Find  the  distance  from  the  origin  to  the  plane  2x  +  2?/  —  0  =  6. 

3.  Find  the  distances  of  the  following  planes  from  the  origin  : 

(a)3a'-4y  +  50-8  =  O,  {h)  x  +  y+  z  =  {), 

(^c)  2y  -^z  =  S,  (d)  3x-^y  +  6  =  0. 

4.  Find  the  distances  from  the  following  planes  to  the  point 
(2,  1,  -  3)  : 

(a)  3  X  +  5 1/  -  6  0  =  8,       (&)  2x-Sy  -  z  =  0,       (c)  x  +  y  +  z=0. 

5.  Find  the  plane  through  the  point  (4,  8,  1)  which  is  perpendicular 
to  the  radius  vector  of  this  point ;  also  the  parallel  plane  whose  distance 
from  the  origin  is  10  and  in  the  same  sense. 

6.  Find  the  plane  through  the  point  (—  1,  2,  —  4)  that  is  parallel  to 
the  plane  ix  —  Sy  +  2z  =  8;  what  is  the  distance  between  these  planes  ? 

7.  Find  the  distance  between  the  planes  4x  —  5?/  —  2^  =  6,  4lX  —  6y 

-20  +  8  =  0. 

8.  Are  the  points  (6,  1,  —  4)  and  (4,  —  2,  3)  on  the  same  side  of  the 
plane  2x  +  Sy  -  5z  +  1  =0? 

9.  Write  down  the  equation  of  the  plane  equally  inclined  to  the  axes 
and  at  the  distance  p  from  the  origin. 

10.  Show  that  the  relation  between  the  distance  p  from  the  origin  to  a 
plane  and  the  intercepts  a,  &,  c  is  1/a^  +  1/b^  4-  1/c^  =  1/p^. 

11.  Show  that  the  locus  of  the  points  equally  distant  from  the  points 
Pi(xi,  ?/i,  0i)  and  Po^x-y,  2/2,  ^2)  is  a  plane  that  bisects  P1P2  at  right 
angles. 

12.  Find  the  equations  of  the  planes  bisecting  the  angles:  (a)  between 
the  planes  x  +  y  -{-  z-3=0,  2x-3i/H- 4^  +  3  =  0;  (6)  between  the 
planes  2x  —  2y  —  z  =  8,x-\-2y  —  2z  =  6. 


XIV,  §  321] 


THE  PLANE 


301 


321.  Volume  of  a  Tetrahedron.  The  volume  of  the  tetrahe- 
dron whose  vertices  are  the  points  Pi(x^,  yi,  Zy),  Pz^x^,  2/2?  ^2)^ 
^(•^*3)  2/3)  h))  Pii^if  2/4)  ^a)  can  be  expressed  in  terms  of  the 
coordinates  of  the  points.  The  equation  of  the  plane  deter- 
mined by  the  points  P2 ,  P3 ,  P4  is  (§  315) 

X     y      z      1 

^2     2/2     Z2     1 

•^3   y^   ^3    1 
^4   yi   z,   1 

Now  the  altitude  d  of  the  tetrahedron  is  the  distance  from  this 
plane  to  the  point  P^  (x^ ,  y^ ,  z^),  i.e.  (§  318) 

2/1     ^i     1 


=  0. 


2/2       ^2        1 

2 

2^2        ^2        1 

2 

«2       2/2       1 

yz   Zz   1 

+ 

2:3        X3        1 

+ 

a^3       2/3        1 

2/4   2:4    1 

Z,      X,       1 

^4       2/4       1 

(^= 


But  the  denominator  is  seen  immediately  to  represent  twice 
the  area  of  the  triangle  with  vertices  P^,  P^,  P^  (Ex.  9,  p.  291), 
i.e.  twice  the  base  of  the  tetrahedron.  Denoting  the  base  by  jB, 
we  then  have 

Xl         2/1         Zy         1 

^'2       2/2       ^2       1 
^Z       2/3       2!3       1 

X,    2/4     2:4     1 
The  volume  of  the  tetrahedron  is  V=  ^Bd,  and  therefore 

Xy     2/1     ^1     1 
2    2/2    '^2    -^ 

•^3       2/3        2!3        1 
X,       2/4       ^4       1 


2Bd 


302 


SOLID  ANALYTIC  GEOMETRY       [XIV,  §  322 


322.   Simultaneous  Linear  Equations.     Two  simultaneous 
equations  of  the  first  degree, 

A^  +  B^y  +  (7,2  -h  A  =  0, 
represent  in  general  the  line  of  intersection  of  the  two  planes 
represented  by  the  two  equations  separately.     For,  the  coordi- 
nates of  every  point  of  this  line,  and  those  of  no  other  point, 
satisfy  both  equations.     See  §  310  and  §§  326-327. 
Three  simultaneous  equations  of  the  first  degree, 

A,x-{-B,y-\-G,z  +  D,  =  0, 

A^  +  B^-\-  C.Z  +  A  =  0, 

A,x  +  B,y  +  Csz  +  A  =  0, 
determine  in  general  the  point  of  intersection  of  the  three 
planes.  The  coordinates  of  this  point  are  found  by  solving 
the  three  equations  for  x,  y,  z.  But  it  may  happen  that  the 
three  planes  have  no  common  point,  as  when  the  three  lines  of 
intersection  are  parallel,  or  when  the  three  planes  are  parallel ; 
and  it  may  happen  that  the  planes  have  an  infinite  number  of 
points  in  common,  as  when  two  of  the  planes,  or  all  three, 
coincide,  or  when  the  three  planes  pass  through  one  and  the 
same  line. 

Four  planes  will  in  general  have  no  point  in  common.     If  they  do,  i.e. 
if  there  exists  a  point  (xi ,  yi ,  zi)  satisfying  the  four  equations 
Aixi  +  Bm  +  C\zi  +  i)i  =  0, 
A2X1  +  BiVi  +  02^1  +  Z)2  =  0, 
Asxi  +  Bm  +  GzZi  +  Z>3  =  0, 
A^Xi  +  BaVi  +  C4Z1  +  2)4  =  0, 

1  between  these  equations  so  that  we  find 


=  0. 


can  eliminate  xi ,  yi, 

^u 

1  between  th 

condition 

A, 

Bi     Ci     Di 

A2 

B2     C2    D2 

As 

Bs     Cs     D, 

A, 

B,     C4     2>4 

XIV,  §323]  THE  PLANE  303 

EXERCISES 

1.  Find  the  volume  of  the  tetrahedron  whose  vertices  are  (0,  0,  0), 
(a,  0,0),  (0,  &,  0),  (0,  0,  c). 

2.  Find  the  volumes  of  the  tetrahedra  whose  vertices  are  the  following 

points  : 

(a)   (7,  0,  6),  (3,  2,  1),  (-  1,  0,  4),   (3,  0,  -  2). 

(6)  (3,  0,  1),   (0,  -  8,  2),   (4,  2,  0),   (0,  0,  10). 

(c)  (2,  1,  -  3),  (4,  -  2,  1),  (3,  -7,  -  4),  (5,  1,  8). 

3.  Find  the  coordinates  of  the  points  in  which  the  following  planes 
intersect : 

(a)  2x  +  67J  -^  z  ~2  =  0,  x  +  6y  +  z  =  0,  3x— 3?/  +  2^— 12=0. 

(6)  2x+y+z=a  +  b  +  c,  ix—2  y-^z=2  a-2b  +  c,  6x~y=Sa-b. 
"^  4.    Show  that   the   four   planes    6x  —  3y  —  z  =  0,   4:X  — 2y-\-z  =  S, 
Sx  +  2y  —  6z  =  6,  x  -{-  y  +  z  =  6  pass  through  the  same  point.     What 
are  the  coordinates  of  this  point  ? 

~  -  5.    Show  that  the  four  planes  4:X  +  y-\-z  +  4:  =  0,  x-\-2y  —  z  +  S=0, 
y— 6z  + li  =  0,  x  +  y  +  z  —  2  =  0  have  a  common  point. 

6.  Show  that  the  locus  of  a  point  the  sura  of  whose  distances  from 
any  number  of  fixed  planes  is  constant  is  a  plane. 

323.  Pencil  of  Planes.  All  the  planes  that  pass  through 
one  and  the  same  line  are  said  to  form  a  pencil  of  planes,  and 
their  common  line  is  called  the  axis  of  the  pencil. 

If  the  equations  of  any  two  non-parallel  planes  are  given, 
say 

A,x  +  B,y  +  C,z  -h  A  =  0, 
A2X  +  ^22/  +  O2Z  4-  A  =  0, 
then  the  equation  of  any  other  plane  of  the  pencil  having  their 
intersection  as  axis  can  be  written  in  the  form 
(2)     {A,x  +  B,y  +  G,z  +  A)  +  A:(^2aj  +  A2/  +  O^z  +  A)  =  0, 
where  /c  is  a  constant  whose  value  determines  the  position  of 
the  plane  in  the  pencil. 

For,  this  equation  (2)  being  of  the  first  degree  in  x,  y,  z 
certainly  represents  a  plane ;  and  the  coordinates  of  the  points 


304  SOLID  ANALYTIC  GEOMETRY       [XIV,  §  323 

of  the  line  of  intersection  of  the  two  given  planes  (1),  since 
they  satisfy  each  of  the  equations  (1),  must  satisfy  the  equa- 
tion (2)  so  that  the  plane  (2)  passes  through  the  axis  of  the 
pencil. 

324.  Sheaf  of  Planes.  All  the  planes  that  pass  through 
one  and  the  same  point  are  said  to  form  a  sheaf  of  planes,  and 
their  common  point  is  called  the  center  of  the  sheaf. 

If  the  equations  of  any  three  planes,  not  of  the  same  pencil, 
are  given,  say 

A^x  +  ^22/+  022;  +  A  =  0, 
A,x-\-B,y-\-C,z-\-D,  =  0, 

then  the  equation  of  any  other  plane  of  the  sheaf  having  their 
point  of  intersection  as  center  can  be  written  in  the  form 
(A,x  +  B,y  +  C,z  4-  A)  +  ^i  (A,x  +  B,y  +  C,z  +  A) 

+  k,  {A,x  +  B^  4-  C,z  +  A)  =  0, 
where   ki  and  k2  are  constants    whose  values   determine   the 
position  of  the  plane  in  the  sheaf. 

The  proof  is  similar  to  that  of  §  323. 

* 

325.  Non-linear  Equations  Representing  Several  Planes. 

When  two  planes  are  given,  say 

A,x  +  B,y  +  C,z-^Di  =  0, 

A^  +  B^  +  Cz  +  D^^O, 
then  the  equation 

(A,x  +  B,y  -f  C,z  -f-  D,)(A,x  +  B,y  +  0,z  -f  A)  =  0, 
obtained  by  equating  to  zero  the  product  of  the  left-hand  mem- 
bers (the  right-hand  members  being  reduced  to  zero),  is  satis- 
fied by  all  the  points  of  the  first  given  plane  as  well  as  all  the 
points  of  the  second  given  plane,  and  by  no  other  points. 

The  product  equation  is  therefore  said  to  represent  the  two 
given  planes.     The  equation  is  of  the  second  degree. 


XIV,  §325]  THE  PLANE  305 

Similarly,  by  equating  to  zero  the  product  of  the  left-hand 
members  of  the  equations  of  three  or  more  planes  (the  right- 
hand  members  being  zero)  we  obtain  a  single  equation  repre- 
senting all  these  planes.  An  equation  of  the  nth  degree  may, 
therefore,  represent  n  planes ;  it  will  do  so  if  its  left-hand  mem- 
ber can  be  resolved  into  n  linear  factors  with  real  coefficients. 

EXERCISES 

1.  Find  the  plane  that  passes  through  the  line  of  intersection  of  the 
planes   5x  —  32/4-4^  —  35=0,    x  +  y  —  z  -.-O   and  through  (4,  —  3,  2) . 

—      2.    Show  that  the  planes  3x  —  22/  +  5  0  +  2=0,  ic  +  ?/  —  0  —  5  =  0, 
6a;  +  2/  +  20—  13  =  0  belong  to  the  same  pencil. 

3.  Show  that  the  following  planes  belong  to  the  same  sheaf  and  find 
the  coordinates  of  the  center  of  the  sheaf :  6a;  +  y  —  42r  =  0,  x  +  |/  +  0  =  5, 
2x  —  4:y-z  =  10,2x  +  Sy+z  =  4. 

4.  What  planes  are  represented  by  the  following  equations  ? 

(a)  a;2-6x  +  8  =  0,  (&)  ^2_9  =  o,  (c)   x'^  -  z^  =  0,  {d)  x'^-4xy  =  0. 

5.  Find  the  cosine  of  the  angle  between  the  following  pairs  of  planes : 
(a)  4:X-Sy-z=6,  x-\-y~z=8  ;  (6)  2x4-7  ^+4^=2,  x-9y-2  0=12. 

6.  Show  that  the  following  pairs  of  planes  are  either  parallel  or 
perpendicular : 

(a)  Sx-2y  +  6z=0,2x+Sy=S;  (b)  6x+2y-z=6,  lOx+iy-2  e=S; 
(c)  x  +  y-2z  =  S,x+y-\-z  =  ll;  (d)  x- 2y  -  z  =  S,  Sx -6y-S  z=6. 

7.  Find  the  plane  that  is  perpendicular  to  the  segment  joining  the 
points  (3,  —  4,  6)  and  (2,  1,  —  3)  at  its  midpoint. 

8.  Show  that  the  planes  Aix  +  Biy  +  Ciz  +  Di=0,  Aix  +  B^y  +  C^z 
-f-jD2  =  0  are  parallel  (on  the  same  or  opposite  sides  of  the  origin)  if 

AxA2  +  BxB2+CiC2      '        _^^ 


VAi^  +  J?i2  +  0/  VA2^  +  B2^  +  CV 

9.  A  cube  whose  edges  have  the  length  a  is  referred  to  a  coordinate 
trihedral,  the  origin  being  taken  at  the  center  of  a  face  and  the  axes  par- 
allel to  the  edges  of  the  cube.     Find  the  equations  of  the  faces. 


306 


SOLID  ANALYTIC  GEOMETRY       [XIV,  §  325 


X 

y 

z 

1 

Xx 

2/1 

Zx 

1 

X2 

2/2 

02 

1 

A 

B 

G 

0 

10.  Show  that  the  plane  through  the  points  Pi(xi,  yi ,  z{)  and 
^1  (Xi,  1/2,  Z2)  and  perpendicular  to  the  plane  Ax -^  By  -\-  Cz  +  D  =  0 
can  be  represented  by  the  equation 


=  0. 


11.  Find  those  planes  of  the  pencil  4a;  —  3^  +  5^  =  8,  2x  +  Sy  —  z  =  4: 
which  are  perpendicular  to  the  coordinate  planes. 

12.  Find  the  plane  that  is  perpendicular  to  the  plane  2x  +  Sy  —  z  =  l 
and  passes  through  the  points  (1,  1,  —  1),  (3,  4,  2). 

13.  Find  the  plane  that  is  perpendicular  to  the  planes  4a;  —  3^  +  0  =  6, 
2aj  +  3?/  —  5^=4  and  passes  through  the  point  (4,  —1,5). 

14.  Show  that  the  conditions  that  three  planes  AiX  +  Biy  +  Ciz  +  Z)i =0, 
A2X  +  B2y  +  C2Z  +  i>2  ==  0,  AzX  +  Bay  +  C^z  +  D3  =  0  belong  to  the  same 
pencil,  are 

Ai+k  A2  _  Bi  +  kB2  _  Ci+kC2  _  D\  +  k  D2 . 
As  B3  O3  2>3 

or,  putting  these  fractions  equal  to  s  and  eliminating  k  and  s, 


B, 

Ci 

2>i 

B2 

O2 

D2 

= 

B, 

O3 

2>3 

D2 

2>3 


Ax        Di    Ax  Bx  Ax  Bx  Cx 

A2  =  D2    A2  B2  =  A2  B2  O2   =0. 

A3        Dz    A3  B3  A3  B3  C3 
(Verify  Ex.  2  by  using  these  conditions. ) 

15.  Find  the  equations  of  the  faces  of  a  right  pyramid,  with  square 
base  of  side  2  a  and  with  altitude  h,  the  origin  being  taken  at  the  center 
of  the  base,  the  axis  Oz  through  the  opposite  vertex  and  the  axes  Ox,  Oy 
parallel  to  the  sides  of  the  base. 

16.  Homogeneous  substances  passing  from  a  liquid  to  a  solid  state  tend 
to  form  crystals  ;  e.g.  an  ideal  specimen  of  ammonium  alum  has  the  form 
of  a  regular  octahedron.  Find  the  equations  of  the  faces  of  such  a  crystal 
of  edge  a  if  the  origin  is  taken  at  the  center  and  the  axes  through  the 
vertices,  and  determine  the  angle  between  two  faces. 

17.  Find  the  angles  between  the  lateral  faces  of  a  right  pyramid  whose 
base  is  a  regular  hexagon  of  side  a  and  whose  altitude  is  h. 


XIV,  §  327] 


THE  STRAIGHT  LINE 


307 


PART   II.     THE   STRAIGHT   LINE 

326.  Determination  of  Direction  Cosines.  Two  simulta- 
neous linear  equations  (§  322), 

(1)  AiJC  +  By+Cz-\-I)=0,   A'3c+B'ij+C'z-{-I)'=0, 

represent  a  line,  namely,  the  intersection  of  the  two  planes 
represented  by  the  two  equations  separately,  provided  the  two 
planes  are  not  parallel. 

To  obtain  the  direction  cosines  I,  m,  n  of  this  line  observe 
that  the  line,  since  it  lies  in  each  of  the  two  planes,  is  perpen- 
dicular to  the  normal  of  each  plane.  Now,  by  §  317  the  direc- 
tion cosines  of  these  normals  are  proportional  to  A,  B,  C  and 
A',  B',  C,  respectively.     We  have  therefore 

Al  +  Bm-j-Cn  =  0,   A'l  -j-  B'm  -{-  C'n  =  0, 
whence 


l:m:7i  = 


BC 
B'C'l 


CA 

C'A' 


AB 
A'B' 


The  direction  cosines  themselves  are  then  found  by  dividing 
each  of  these  determinants  by  the  square  root  of  the  sum  of 
their  squares. 

327.   Intersecting  Lines.     The  two  lines 


A^x  -f  B,y  +  C,z  +  A  =  0,  ] 
A,'x-\-B,'y-rC,'z-}-D,'=0  J 


I  A^'x+B^'y^  C^z-^D^  =  0 


will  intersect  if,  and  only  if,  the  four  planes  represented  by 

these  equations  have  a  common  point.     By  §  322,  the  condition 

for  this  is 

A^    B,    C,   A 

A,'  B,'  C  A 

A2       X>2       C2      -U2 

A^  Bi  C^  A'  "'I' 


=  0. 


308  SOLID  ANALYTIC  GEOMETRY       [XIV,  §  328 

328.  Special  Forms  of  Equations.  For  many  purposes  it  is 
convenient  to  represent  a  line  by  means  of  one  of  its  points 
and  its  direction  cosines,  or  by  means  of  two  of  its  points. 
Let  the  line  be  called  X. 

If  (^a,  2/i>  %)  is  a  given  point  of  A.  and  I,  m,  n  are  the  direc- 
tion cosines  of  A,  then  every  point  (x,  y,  z)  of  A.  must  satisfy 
the  relations  (§  298)  : 

^  ^  I  m  n      ' 

In  these  equations,  Z,  m,  n,  can  evidently  he  replaced  by  any 
three  numbers  proportional  to  I,  m,  n.  Thus,  if  (^2,  y2,  z.,)  be 
any  point  of  A,  different  from  (a^,  2/1,  ^j),  we  have  the  continued 
proportion 

0^2  —  ajj :  2/2  —  2/1 : 2!2  —  2i  =  Z :  m  :  n ; 

hence  the  equations  of  the  line  through  the  two  points  (x^ ,  2/1  j  ^1) 
and  (X2 , 2/2 ,  Z2)  are : 

(3)  00-iCt  ^  y-Vi  ^  g-^i  ^ 

i»2-«i      2/2-2/1      «2-«i* 

If,  for  the  sake  of  brevity,  we  put  x^—  x^  =  a,  2/2  —  2/i  =  ^> 
Z2  —  Zi=:  c,  we  can  write  the  equations  of  the  line  in  the  form 

^  a  b  c     ' 

where  a,  b,  c,  are  proportional  to  I,  m,  n,  and  can  be  regarded  as 
the  components  of  a  vector  parallel  to  the  line. 

The  equations  (3)  also  follow  directly  by  eliminating  k  be- 
tween the  equations  of  §  295,  namely, 

(5)    a?=a?i-f-A;(a?2-a5i),  y=y^^k{y^-y{),  z=z^-]-Jc(z^-z{). 

These  equations  which,  with  a  variable  h,  represent  any  point 
of  the  line  through  (iCi ,  2/1 ,  ^j)  and  {x^ ,  2/2  >  ^2)  are  called  the 
parameter  equations  of  the  line. 


XIV,  §  329] 


THE  STRAIGHT   LINE 


309 


329.  Projecting  Planes  of  a  Line.  Each  of  the  forms  (2), 
(3),  (4),  which  are  not  essentially  different,  furnishes  three 
linear  equations ;  thus  (4)  gives  : 


y 


c 


a 


Vi 


he  c  a  ah 

but  these  three  equations  are  equivalent  to  only  two,  since  from 
any  two  the  third  follows  immediately. 
The    first   of   these   equations,   which 
can  be  written  in  the  form 

cy-hz-{cyy-hz;)=0, 

represents,  since  it  does  not  contain  x 
(§  313),  a  plane  perpendicular  to  the 
plane  0?/2;  and  as  this  plane  must  con- 
tain the  line  X  it  is  the  plane  CCA 
that  projects  \  on  the  plane  Oyz  (Fig.  135).  Similarly  the  other 
two  equations  represent  the  planes  that  project  \  on  the  co- 
ordinate planes  Ozx  and  Oxy.  Any  two  of  these  equations 
represent  the  line  X  as  the  intersection  of  two  of  these  pro- 
jecting planes. 

At  the  same  time  the  equation 


Fig.  135 


can  be  interpreted  as  representing  a  line  in  the  plane  Oyz, 
viz.  the  intersection  of  the  projecting  plane  with  the  plane 
£C  =  0.  This  line  {AC  in  Fig.  135)  is  the  projection  X^  of  X  on 
the  plane  Oyz.  As  the  other  two  equations  (4)  can  be  inter- 
preted similarly  it  appears  that  the  equations  (2),  (3),  or  (4) 
represent  the  line  X  by  means  of  its  projections  A^.,  X^,  A,  on 
the  three  coordinate  planes,  just  as  is  done  in  descriptive 
geometry.  Any  two  of  the  projections  are  of  course  sufficient 
to  determine  the  line. 


310  SOLID  ANALYTIC  GEOMETRY       [XIV,  §  330 

330.  Determination  of  Projecting  Planes.  To  reduce  the 
equations  of  a  line  A  given  in  the  form  (1)  to  the  form  (4)  we 
have  only  to  eliminate  between  the  equations  (1)  first  one  of 
the  variables  x,  y,  z,  then  another,  so  as  to  obtain  two  equa- 
tions, each  in  only  two  variables  (not  the  same  in  both). 

The  process  will  best  be  understood  from  an  example.  The 
line  being  given  as  the  intersection  of  the  planes 

(a)  2x-3y-\-z  +  3=:0, 
{b)  x-\-y-j-z-2  =  0, 

eliminate  z  by  subtracting  (b)  from  (a)  and  eliminate  x  by 
subtracting  (6),  multiplied  by  2,  from  (a) ;  this  gives  the  line 
as  the  intersection  of  the  planes 

x  —  4:y  -\-5  =  0j 
-5y-^z  +  7  =  0, 

which  are  the  projecting  planes  parallel  to  Oz  and  Ox,  i.e.  the 
planes  that  project  the  line  on  Oxy  and  Oyz.  Solving  for  y 
and  equating  the  two  values  of  y  we  find : 

x-\-5  _y  _z  —  T 
4    "1""  -5* 

The  line  passes  therefore  through  the  point  (—5,  0,  7)  and 
has  direction  cosines  proportional  to  4,  1,  —  5,  viz. 

,4  1  5 

I  — ,         m 


V42  V42  V42 

EXERCISES 

1.  Write  the  equations  of  the  line  through  the  point  (—  3, 1,  6)  whose 
direction  cosines  are  proportional  to  3,  5,  7. 

2.  Write  the  equations  of  the  line  through  the  point  (3,  2  —  4)  whose 
direction  cosines  are  proportional  to  5,  —  1,  3. 

3.  Find  the  line  through  the  point  (a,  6,  c)  that  is  equally  inclined 
to  the  axes  of  coordinates. 


XIV,  §  331]  THE  STRAIGHT  LINE  311 

4.  Find  the  lines  that  pass  through  the  following  pairs  of  points : 
.    (a)  (4,  -  3,  1),  (2,  3,  2),  (&)  (-  1,  2,  3),  (8,  7,  1), 

(c)  (-2,3,  -4),  (0,2,0),  (d)   (-1,  -5,  -2),,  (-3,0,-1), 

and  determine  the  direction  cosines  of  each  of  these  lines. 

5.  Find  the  traces  of  the  plane  2  x  —  3  «/  -  4  ^  =  6  in  the  coordinate 
planes. 

^      6.   Write  the  equations  of  the\me2x-Sy  +  5  z~6=^0,x—y+2z-S=0 
in  the  form  (4)  and  determine  the  direction  cosines. 

7.  Put  the  line  4:X  — Sy  — 6  =  0,  x-y-z-4:  =  0  in  the  form  (4) 
and  determine  the  direction  cosines. 

8.  Find  the  line  through  the  point  (2,  1,-3)  that  is  parallel  to  the 
]ine2x-Sy-\-4z-6  =  0,  6x  +  y-2z-S  =  0. 

9.  What  are  the  projections  of  the  line  5x  —  3?/  —  7^;  — 10  =  0, 
X  -\-y  —  S z  +6  =  0  on  the  coordinate  planes ? 

10.  Obtain  the  equations  of  the  line  through  two  given  points  by- 
equating  the  values  of  k  obtained  from  §  295. 

11.  By  §  317,  the  direction  cosines  of  any  line  are  proportional  to  the 
coefficients  of  x,  y,  and  z  in  the  equation  of  a  plane  perpendicular  to  the 
line.  Find  a  line  through  the  point  (3,  5,  8)  that  is  perpendicular  to  the 
plane  2x  +  i/  +  30  =  5. 

331.  Angle  between  Two  Lines.  The  cosine  of  the  angle  ^p  be- 
tween two  lines  whose  direction  cosines  are  h,  mi,  wi  and  h,  rrn,  nz  is, 

by  §  299, 

cos  \p  =  hh  +  W2im2  +  wiW2. 

Hence  if  the  lines  are  given  in  the  form  (4) ,  say 

x-xi  _  y  —  yi  _  z  -  zi     x-xi  _  y  —  yt  _  z  —  zt 
«i  &i  c'l  ai  62  C2 

we  have 

cos  V  = .  ^^^^  "^  ^^^2  "•"  ^^^2 


±  Vai2  +  &i2  +  ci2  .  ±  Va22  +  62^  +  C22 

If  the  lines  are  parallel,  then 

ai_  &i  _ci. 
a2,     02     C2 
if  they  are  perpendicular,  then 

aia2  +  61&2  +  C1C2  =  0  ; 
and  vice  versa* 


312 


SOLID  ANALYTIC  GEOMETRY       [XIV,  §  332 


Let  the  line  and  plane 


332.   Angle  between  Line  and  Plane. 

be  given  by  the  equations 

x  —  xi_y-  y\_z  —  zi 
a  h  c     ' 

Ax-\-  By  +  Cz-\r  D  =  (i. 

The  plane  of  Fig.  136  represents  the  plane 
through  the  given  line  perpendicular  to  the  given 
plane.    The  angle  /3  between  the  given  line  and 
plane  is  the  complement  of  the  angle  a  between  the  line  and  any  perpen- 
dicular PiV  to  the  plane.     Hence 

.„«_  aA-^hB-\-cC 


Fig.  136 


±    v/a2  +  6-2  +  C2  .  ±    V^2  +   ^2  +   (72 

The  (necessary  and  sufficient)  condition  for  parallelism  of  line  and 

plane  is 

aA  +  hB  +  cC  =  0\ 

the  condition  of  perpendicularity  is 

a_  _h  _  c^ 
A~  B~  G 

333.  Line  and  Plane  Perpendicular  at  Given  Point.    If  the 

plana  Ax  +  By  -{■  Cz -{■  D  =  Q 

passes  through  the  point  Pi(xi ,  yi ,  zi),  we  must  have 
Axi  +  Byx  +  Czx  +  D  =  ^. 
Subtracting  from  the  preceding  equation,  we  have  as  the  equation  of 
any  plane  through  the  point  Pi(xi,  yi,  z\)  : 

A(x  -  xi)  +  Biy  -  yx)  +  C{z  -  zi)  =  0. 
The  equations  of  any  line  through  the  same  point  are 

x  —  xi  _y  -yi  _z  —  zi^ 
a  b  c 

If  this  line  is  perpendicular  to  the  plane,  we  must  have  (§  332)  :  a/ A  = 
b/B  =  c/C.     Hence  the  equations 

x  —  xi_y—yi_z  —  zi 


represent  the   line  through  Pi(iCi,  yi,  zi)    perpendicular  to  the  plane 
Aix  -  xi)  +  Biy  -  2/0  +  C{z  -  zi)  =  0. 


XIV,  §  335] 


THE  STRAIGHT  LINE 


313 


If  the  equations  of 


Fig.  137 


334.  Distance  of  a  Point  from  a  Line. 

the  line  X  are  given  in  the  form 

x—xi  _ y  —  yi  _ z  —  zi " 
I  m  n 

where  {xi ,  yi ,  z\)  is  a  point  Pi  of  X  (Fig. 
187),  the  distance  d  =  QP2  of  the  point 
•P2(a;2,  ^2,  Z2)  from  X  can  be  found  from 
the  right-angled  triangle  Pi  QP2  which  gives 

cP  =  FiP2^  -  PiQ^, 
by  observing  that 

P1P22  =  (X2  -  XiY  +  (2/2  -yi)2  +  (Z2  -  Zi)\ 

while  PiQ  is  the  projection  of  P1P2  on  X.  This  projection  is  found 
(§  294)  as  the  sum  of  the  projections  of  the  components  X2  —  xu  y'z  —  yu 
Z2  —  z\  of  P1P2  on  X  : 

PiQ  =  1(X2  —  xi)  +  m(y2  -  y\)  +  n{z2  -  z{). 
Hence 

(?2=:(a;2-xi)2+  (y2-2/i)2  +  (^2-^i)2-[Z(a;2-xi)+m(?/2-yi)  +n{z2-z{)Y' 

335.  Shortest  Distance    between   Two  Lines.     Two  lines 

Xi ,  X2  whose  equations  are  given  in  the  form 

-  2/1  _  Z—Zi       X-X2  _  y  —  y2  _  Z-  Z2 


X  —  X\ 

h 


mi 


W2 


W2 


will  intersect  if  their  directions  {h ,  mi ,  ni),  (I2-,  wi2,  n2),  and  the  direc- 
tion of  the  line  joining  the  points  {x\ ,  yi ,  z{)^  (x.2 ,  2/2 ,  ^2)  are  complanar 
(§302),  i.e.  if 

X2  —  Xi      2/2  -  2/1      Z2  -  Z\ 

l\  m\  n\ 

I2  WI2  W2 

If  the  lines  Xi ,  X2  do  not  intersect,  their  shortest  distance  d  is  the  dis- 
tance of  P2(aj2,  ^2,  Z2)  from  the  plane  through  Xi  parallel  to  X2.  As  this 
plane  contains  the  directions  of  Xi  and  X2 ,  the  direction  cosines  of  its  nor- 
mal are  (§  301)  proportional  to 


mi 

ni 

ni     h 

h    mi 

m2 

«2 

» 

n2    h 

5 

I2    m2 

314 


SOLID  ANALYTIC  GEOMETRY       [XIV,  §  335 


and  as  it  passes  through  Pi  {xi ,  yi ,  ^i)  its  equation  can  be  written  in  the 
form 

x  —  xi    y  —  yiz  —  zi 

h  wii  wi      =  0. 

h  Wl2  W2 

Hence  the  shortest  distance  of  the  lines  Xi,  X2  is  : 


d  = 


V 


X2  -xi    y2-  yi 

Z2- 

-^1 

h                  Wli                 Wi 

h            m2            W2 

n 
n 

il      Wi 
l2      W2 

2 

+ 

ni     Zi 
W2     h 

2 

+ 

h 
h 

mi 
m2 

As  the  denominator  of  this  expression  is  equal  to  sirn/'  (§  301),  we  have 

X2  —  Xi      ?/2  —  2/1      2;2  -  ^1 

d  sin  xp  =        li  mi  ui 

I2  Wl2  W2 


EXERCISES 
— '"     1.   Find  the  cosine  of  the  anojle  between  the  lines 

X 


^-yj:zA-^±l^  ^ 


l_y -3_g  +  3 
-1  2  3 


2  3  4 

2.  Find    the    angle    between    the    lines    3x  —  2?/  +  42  —  1  =  0, 
2x  +  y— 3^  +  10  =  0,  and  x-\-y  -\-  z  =  Q,  2x  +  3y-5;s  =  8. 

3.  Find  the  angle  between  the  lines  that  pass  through  the  points 
(4,  2,  5),  (-  2,  4,  3)  and  (-  1,  4,  2),  (4,  -  2,  -  6). 

4.  Find  the  angle  between  the  line 

a;  +  l  _y-2  _g+  10 
3  -5  3 

and  a  perpendicular  to  the  plane  4a:  —  3?/  —  2^  =  8. 

6.   In  what  ratio  does  the  plane  3x— 4i/  +  65;  —  8  =  0   divide   the 
segment  drawn  from  the  origin  to  the  point  (10,  —  8,  4). 

6.   Find  the  plane  through  the  point  (2,  —  1,  3)  perpendicular  to  the 


line 


x  —  '6     y  +  2  _z  —  7 


XIV,  §335]  THE  STRAIGHT   LINE  315 

7.    Find  the  plane  that  is  perpendicular  to  the  line  ^x-{-y  —  z=6, 
3x  +  4?/  +  82+  10  =0  and  passes  through  the  point  (4,  —1,3), 

*—      8.    Find  the  plane  through  the  origin  perpendicular  to  the  line 
Sx-2y  +  z=6,   Sx  +  y -4z  =  S. 

9.    Find  the  plane  through  the  point  (4,  —  3,  1)  perpendicular  to  the 

line  joining  the  points  (3,  1,  —  6),  (—  2,  4,  7). 

10.    Find  the  line  through  the  point  (2,  —  1,  4)  perpendicular  to  the 
plane  x  —  2y-\-4:Z  =  6. 
■""      11.    Show  that  the  lines  x/S  =  y/  —1  =  z/—2  and  x/4:  —  y/6  =  z/S  are 
perpendicular. 

—  12.    Show  that  the  lines 

^izi  =  L+2^£j::^  ^^^  x-2^y-S^   z 
1-2  3  _2  4  -6 

are  parallel. 

—  13.   Find  the  angle  between  the  line  3  ic  —  2  y  —  ^  =  4,  4  a;  +  3  ?/ —  3  ^  =  6 
and  the  plane  x  -\-y  -\-  z  —  %. 

14.  Find  the  lines  bisecting  the  angles  between  the  lines 

X—  O'  -V  -  h  _z  —  c  ^^^^  x  —  a__y—  h  _z  —  c 
h  nil  ni  h  m^  nt 

15.  Find  the  plane  perpendicular  to  the  plane  Zx  —  ^y  —  z  —  Q  and 
passing  through  the  points  (1,  3,  —  2),  (2,  1,  4). 

16.  Find  the  plane  through  the  point  (3,  —  1,  2)  perpendicular  to  the 
line  2x  — 3y  —  42;  =  7,   x-\-  y  —  2z  =  ^. 

17.  Find  the  plane  through  the  point  (a,  6,  c)  perpendicular  to  the 
line  Axx  +  Bxy  +  Cxz  +  Z)i  =  0,   Aix  +  Bty  +  Ciz  +  Z>2  =  0. 

18.  Find  the  projection  of  the  vector  from  (3,  4,  5)  to  (2,  —  1,  4)  on  the 
line  that  makes  equal  angles  with  the  axes  ;  and  on  the  plane 

2x-3?/  +  4^=6. 

19.  Find  the  distances  from  the  following  lines  to  the  points  indicated : 

^""^  1  =  ^  =  4^'   (0,0,0); 

(6)  2x  +  y-5r  =  6,  a:-?/  +  4^  =  8,  (3,  1,4); 

(c)  2a;  +  32/  +  50  =  l,  3x-6?/  +  3;?=0,  (4,  1,  -2). 


316  SOLID  ANALYTIC  GEOMETRY       [XIV,  §  335 

20.   Show  that  the  equation  of  the  plane  determined  by  the  line 

x  —  xi_y  —  yi_z  —  zi 
a  b  c 

and  the  point  P2  (xz ,  1/2 ,  ^2)  can  be  written  in  the  form 

0. 


X  -xi    y  —yi    z  -z\ 

xi  -x\    yi  —  7/1    zi  —  zi 

a  b  c 


21.  Find  the  plane  determined  by  the  intersecting  lines 

x-3^y-6^z  +  l  and'^~^  =  y~^  =  ^  +  ^ 
4  3  2  1  2  3     * 

22.  Find  the  plane  determined  by  the  line 

x-xi  _  y  —  yi  _  z  ~  zi 
a  b  c     ^ 

and  its  parallel  through  the  point  P2  (X2 ,  2/2 ,  ^2). 

23.  Given  two  non-intersecting  lines 

x  —  xi  _  y  —  yi  _  z  -  z\     x  —  xi  _  y  -yi  _  z  —  zi . 
a\  b\  c\  at  62  C2 

find  the  plane  passing  through  the  first  line  and  a  parallel  to  the  second  ; 
and  the  plane  passing  through  the  second  line  and  a  parallel  to  the  first. 

24.  What  is  the  condition  that  the  two  lines  of  Ex.  23  intersect  ? 

25.  Find  the  distance  from  the  diagonal  of  a  cube  to  a  vertex  not  on 
the  diagonal. 

26.  Find  the  distance  between  the  lines  given  in  Ex.  23. 

27.  Show  that  the  locus  of  the  points  whose  distances  from  two  fixed 
planes  are  in  constant  ratio  is  a  plane. 

28.  Show  that  the  plane  (w  —  n)x  +  (n  —  Z)?/  +  (Z  —  m)z  =  0  contains 
the  line  x/l  =  y/m  =  z/n  and  is  perpendicular  to  the  plane  determined  by 
the  lines  x/m  =  y/n  =  z/l  and  x/n  =  y/l  =  z/m. 


CHAPTER  XV 

^  THE   SPHERE 

336.  Spheres.  A  sphere  is  defined  as  the  locus  of  all  those 
points  that  have  the  same  distance  from  a  fixed  point. 

Let  CQi,  j,  k)  denote  the  center,  and  ?•  the  radius,  of  a  sphere ; 
the  necessary  and  sufficient  condition  that  any  point  F(x,  y,  z) 
has  the  distance  r  from  C{1i,j,  k)  is 

(1)  {x -  hY  +  {y  -jY  -viz-  ky  =  rK 

This   then   is   the   cartesian   equation   of  the  sphere  of  center 
C(h,  j,  k)  and  radius  r. 

If  the  center  of  the  sphere  lies  in  the  plane  Oxy^  the  equa- 
tion becomes 

(x-hy-\-{y-jy  +  z'=r\ 

If  the  center  lies  on  the  axis  Ox,  the  equation  is 

(x-hf+y^-\-z'^  =  r\ 
The  equation  of  a  sphere  about  the  origin  as  center  is : 

x2  +  i/2  +  s2  =  r2. 

337.  Expanded  Form.  Expanding  the  squares  in  the  equa- 
tion (1),  we  find  the  equation  of  the  sphere  in  the  form 

3(?  +  y^-\-z^-2hx-2jy-2kz-\-h''-^f-{-¥-r''=:0. 

This  is  an  equation  of  the  second  degree  in  x,  y,  z;  but  it  is  of 
a  particular  form. 

The  general  equation  of  the  second  degree  in  x,  y,  z  is 

Ax^  +  By^ -\-Cz^  +  2Dyz-{-2Ezx-\-2  Fxy 

+  2Gx-^2Hy-\-2Iz-^J=0; 
317 


318  SOLID  ANALYTIC  GEOMETRY        [XV,  §  337 

i.e.  it  contains  a  constant  term  J-,    three  terras  of  the  first 
degree,  one  in  x,  one  in  y,  and  one  in  z ;  and  six  terms  of  the 
second  degree,  one  each  in  x^,  y^,  z"^,  yz,  zx,  and  xy. 
If  in  the  general  equation  we  have 

D  =  E  =  F=0,  A  =  B=C=itO, 

it  reduces,  upon  division  by  A,  to  the  form 

x^  +  f  +  z^  +  ^x  +  ^-^y  +  ?^z  +  ^  =  0, 

which  agrees  with  the  above  form  of  the  equation  of  a  sphere, 
apart  from  the  notation  for  the  coefficients. 

338.  Determination  of  Center  and  Radius.     To  determine 
the  locus  represented  by  the  equation 

(2)        Ax-'-\-  Ay^ -^ Az^  +  2  Gx-\-2 By -{-2 Iz  +  J=:0, 

where  A,  G,  Hy  7,  J,  are  any  real  numbers  while  ^  ^^  0,  we 
divide  by  A  and  complete  the  squares  in  x,  y,  z\  this  gives 


(-!J-('-3"+('+3)" 


The  left  side  represents  the  square  of  the  distance  of  the  point 
(x,  y,  z)  from  the  point  (—  Gj A^  —  H/A,  —  I/A) ;  the  right 
side  is  constant.  Hence,  if  the  right  side  is  positive,  the  equa- 
tion represents  the  sphere  whose  center  has  the  coordinates 

and  whose  radius  is 


r^-^G'^^H^  +  P-AJ. 
A 

If,  however,  G"^  -\-  A"^  +  I^  <  A  J,  the  equation  is  not  satisfied  by 

any  point  with  real  coordinates.     If   G^ -\- H^ -{•  I'^  =  AJ,  the 

equation   is   satisfied   only  by  the  coordinates  of   the   point 

{-G/A,-H/A,-I/A). 


XV,  §  340] 


THE  SPHERE 


319 


n 


Thus  the  equation  of  the  second  degree 

Ax^  +  By"^  -f-  C22  +  2  Dyz  +  2  Ezx  +  2  Fxy 

+  2Gx^2Hy-{-2Iz-\-J=0, 

represents  a  sphere  if,  and  orily  if 

A=:B=C^O,     D  =  E  =  F=0,     G^-\-H^-\-P>AJ. 

339.  Essential  Constants.  The  equation  (1)  of  the  sphere 
contains  four  constants  :  h,  j,  k,  r.  The  equation  (2)  contains 
five  constants  of  which,  however,  only  four  are  essential  since 
we  can  divide  out  by  one  of  these  constants.  Thus  dividing 
by  A  and  putting  2  G/A  =  a,  2  H/A  =  h,  2  I/A  =  c,  J/ A  =  d, 
the  general  equation  (2)  assumes  the  form 

x"^  +  y^  +  z"^ -\- ax -{- by -^  cz  +  d  =  Oy 
with  only  the  four  essential  constants  a,  b,  c,  d. 

This  fact  corresponds  to  the  possibility  of  determining  a 
sphere  geometrically,  in  a  variety  of  ways,  by  four  conditions. 

340.  Sphere  through  Four  Points.  To  find  the  equation  of  the 
sphere  passing  through  four  points  Fi(zi,  ?/i,  ^i),  P2(X2,  y^-,  ^2), 
^3(353,  ^3,  ^3),  P^(Xi,  y4,  Z4),  observe  that  the  coordinates  of  these  points 

•  must  satisfy  the  equation  of  the  sphere 

^2  _|_  ^2   ^  ^2  ^(j^x  +by  +CZ  +  d  =  0; 
i.e.  we  must  have 

xi^  +  Vi^  +  z^  +  axx  +  byx  +  csri  +  d  =  0, 
X'^  +  y<^  +  zci^  +  ax2  +  byi  -^czi-^-d-^, 
x-^  4-  yz^  +  z^^  +  axi  +  hyz  +  C23  +  (^  =  0, 
x^  +  y^  +  z^  +  ax4  +  by 4  -\-  cz^^-d-^. 
As  these  five  equations  are  linear  and  homogeneous  in  1,  a,  &,  c,  (?,  we 
can  eliminate  these  five  quantities  by  placing  the  determinant  of  their 
coefficients  equal  to  zero.    Hence  the  equation  of  the  desired  sphere  is 

aj2      ^   y2       _J.   ^2  a;  y  ^         ■ 

x^^y^^z^    xx    y\    zi    1 

X2^  +  y2^  +  Z2^      X2      y2      ^2      1 

Xz^  +  yz^  +  zz^    xz     yz     zz     1 

X4^  +  2/4^  +  Z4^      Xa      2/4      Z4      1 


320  SOLID  ANALYTIC  GEOMETRY        [XV,  §  340 

EXERCISES 

1.  Find  the  spheres  with  the  following  points  as  centers  and  with  the 
indicated  radii : 

"^    (a)   (4,  -1,2),  4;    (6)   (0,0,  4),  4;    (c)   (2,-2,  1),  3;    (d)   (3,  4,  1),  7. 

2.  Find  the  following  spheres : 

-    (a)  with  the  points  (4,  2,  1)  and  (3,  —  7,  4)  as  ends  of  a  diameter  ; 

—  (6)  tangent  to  the  coordinate  planes  and  of  radius  a  ; 

(c)  with  center  at  the  point  (4,  1,  5)  and  passing  through  (8,  3,  —  5). 

3.  Find  the  centers  and  the  radii  of  the  following  spheres  : 
(a)  a;2  +  ?/2  +  ;s2  _  3  X  +  5  y  -  6  ^  +  2  =  0. 

-  (6)  a:2  +  2/2  +  ^2  _  2  6a;  +  2  cs  -  &2  _  c2  =  0. 

(c)  2  x2  +  2  y2  +  2  ^2  ^  3  X  -  y  +  5  0  -  11  =  0. 

(d)  x^  +  y^-\-  z^-x-y  -  z  =  0. 

__      4.    Show  that  the  equation  A{x^  -i- y^  +  z^)  +2  Gx  +  2  Hy  -\- 2  Iz +J 
=  0,  in  which  J  is  variable,  represents  a  family  of  concentric  spheres. 

5.  Find  the  spheres  that  pass  through  the  following  points  : 

—  (a)  (1,  1,  1),  (3,  -  1,  4),  (-  1,  2,  1),  (0,  1,  0). 
(6)  (0,  0,  0),  (a,  0,  0),  (0,  6,  0),  (0,  0,  c). 

(c)  (0,  0,  0),  (-  1,  1,  0),  (1,  0,  2),  (0,  1,  -  1). 

(d)  (0,0,  0),  (0,0,4),  (3,3,3),  (0,4,0). 

6.  Find  the  center  and  radius  of  the  sphere  that  is  the  locus  of  the 
points  three  times  as  far  from  the  point  (a,  6,  c)  as  from  the  origin. 

—  7.    Show  that  the  locus  of  the  points,  the  ratio  of  whose  distances  from 
two  given  points  is  constant,  is  a  sphere  except  when  the  ratio  is  unity. 

—  8.  Find  the  positions  of  the  following  points  relative  to  the  sphere 
jc2  +  ?/2  +  ;s2_4a;  +  4y-2;s  =  0;  (a)  the  origin,  (6)  (2,  -2,  1), 
(c)   (1,1,1),  (d)   C3,  -2,1). 

9.   Find  the  positions  of  the  following  planes  relative  to  the  sphere 
x2  4-«/2+02  +  4x-3?/  +  6«  +  5  =  O: 
(a)  4:X-\-2y  +  z  +  2  =  0,  (b)Sx-y-^z  +  6  =  0. 

10.  Find  the  positions  of  the  following  lines  relative  to  the  sphere  of 
Ex.  9  :  (a)2x-y  +  2z  +  7  =  0,    Sx-    y-z -10  =  0. 

(by  Bx  +  8y  +  z  -9=0,       x-8y-{-z-\-n  =  0. 

11.  Find  the  coordinates  of  the  ends  of  that  diameter  of  the  sphere 
x^  +  y^  +  z'^  —  6x  —  6y-\-4iZ  —  QQ  =  0,  which  lies  on  the  line  joining  the 
origin  and  the  center. 


XV,  §  342]  THE  SPHERE  321 

341.  Equations  of  a  Circle.  In  solid  analytic  geometry  a 
curve  is  represented  by  two  simultaneous  equations  (§  310), 
that  is,  by  the  equations  of  any  two  surfaces  intersecting  in 
the  curve.  Thus  two  linear  equations  represent  together  the 
line  of  intersection  of  the  two  planes  represented  by  the  two 
equations  taken  separately  (§§322,  326). 

A  linear  equation  together  with  the  equation  of  a  sphere, 

^  ^  x^  ■\- y"^  ■\- z^  ^  ax -\- by  -\- cz -\- d  =  (), 

represents  the  locus  of  all  those  points,  and  only  those  points, 
which  the  plane  and  sphere  have  in  common.  Thus,  if  the 
plane  intersects  the  sphere,  these  simultaneous  equations  rep- 
resent the  circle  in  which  the  plane  cuts  the  sphere;  if  the 
plane  is  tangent  to  the  sphere,  the  equations  represent  the 
point  of  contact;  if  the  plane  does  not  intersect  or  touch 
the  sphere,  the  equations  are  not  satisfied  simultaneously  by 
any  real  point. 

342.  Sections  Perpendicular  to  Axes.  Projecting  Cylinders. 

In  particular,  the  simultaneous  equations 

(4)  z  =  Tc,         ic2  +  2/2  +  ;s2  ^  7-2 

represent,  if  A:  <  r,  a  circle  about   the  axis   Oz   {i.e.   a   circle 

whose  center  lies  on  Oz  and  whose  plane  is  perpendicular  to 

Oz).     If  the  value  of  z  obtained  from  the  linear  equation  be 

substituted  in  the  equation  of  the  sphere,  we  obtain  an  equation 

in  X  and  y,  viz.  „        „        „      , « 

which  represents  (since  z  is  arbitrary)  the  circular  cylinder, 
about  Oz  as  axis,  which  projects  the  circle  (4)  on  the  plane 
Oxy.  Interpreted  in  the  plane  Oxy,  i.e.  taken  together  with 
2  =  0,  this  equation  represents  the  projection  of  the  circle  (4) 
on  the  plane  Oxy. 

Similarly  if  we  eliminate  x  ov  y  ot  z  between  the  equations 


322  SOLID  ANALYTIC  GEOMETRY        [XV,  §  342 

(3)  we  obtain  an  equation  in  y  and  z,  z  and  x,  or  x  and  ?/,  rep- 
resenting the  cylinder  that  projects  the  circle  (3)  on  the  plane 
Oyz,  Ozx,  or  Oxy,  respectively. 

343.  Tangent  Plane.  The  tangent  plane  to  a  sphere  at  any 
point  Pi  of  the  sphere  is  the  plane  through  P^  at  right  angles 
to  the  radius  through  Pj . 

For  a  sphere  whose  center  is  at  the  origin, 
^  +  y"^  -\-z'^  =  r^, 
the  equation  of  the  tangent  plane  at  P\{x-^,  yi,  Zi)  is  found  by 
observing  that  its  distance  from  the  origin  is  r  and  that  the 
direction  cosines  of  its  normal  are  those  of   OPi,  viz.  Xi/r, 
yi/r,Zi/r.     Hence  the  equation 

(5)  x^x  +  2/i2/  +  ^iZ  =  r\ 

If  the  equation  of  the  sphere  is  given  in  the  general  form 
A{x^  +y''+z'')+2  0x  +  2Hy  -\-2Iz  ^  J=0, 
we  obtain  by  transforming  to  parallel  axes  through  the  center 
the  equation 

the  tangent  plane  at  P^ix^,  2/1  ?  %)  ^^^^^  is 

a^ia^ +  2/12/4- ^1^  =  ^  +  ^,  +  ^-- 
Transforming  back  to  the  original  axes,  we  have : 

(--S(-S)*('-!)("!)H-3(-i) 

A^      A^     A^     a' 
Multiplying  out  and  rearranging,  we  find  that  the  equation  of 
the  tangent  plane  to  the  sphere 

Aix"  +  y""  +  z'')  +  2  Qx  +  2 Hy  -\-2 Iz  +  J  =  0 

at  the  point  Pi  (fl?i,  2/1?  ^\)  is 

(6)  A{x^x-\-yiy+z^z)^-0(x,^x)  +H{y,^y)+I{z^  +  z)+  J=  0. 


XV,  §  344]  THE  SPHERE  323 

344.   Intersection  of  Line  and  Sphere.     The  intersections 
of  a  sphere  about  the  origin, 

x^  -\-  y^  -{-  z"^  =  r^, 
with  a  line  determined  by  two  of  its  points  Pi(xi,yi,  Zi)  and 
PgC^aj  Vi)  ^2))  and  given  in  the  parameter  form  [(5),  §  328] 

x  =  Xi  +  k{x2-x{),  y  =  yi-\-Jc(y2-yi),  z=z^-\-'k{z2-z{), 
are  found  by  substituting  these  values  of  ic,  ?/,  z  in  the  equation 
of  the  sphere  and  solving  the  resulting  quadratic  equation  in  k : 
[x,  +  J€(x,  -  x,)Y  +  [?A  +  k(y2  -  yi)Y  +  [^1  +  k(z,  -  z{)Y  =  r\ 
which  takes  the  form 
\_(x^  -  a;i)2  -h  {y,  -  y^y  +  (z^  -  z^y^  k^  +  2  [x,  {x^  -  x,)  -\-  y^  {y^  -  y,) 

The  line  P1P2  will  intersect  the  sphere  in 
two  different  points,  be  tangent  to  the 
sphere,  or  not  meet  it  at  all,  according  as 
the  roots  of  this  equation  in  k  are  real  and 
different,  real  and  equal,  or  imaginary ;  i.e. 
according  as 

where  d  denotes  the  distance  of  the  points  Pi  and  Pg.  Divid- 
ing by  d^,  we  can  write  this  condition  in  the  form 

r'  -  \x,^  +  2/1^  +  z,^  -  U""-^'  +  2/1^^'  +  ^i^-^'Y]  I  ^> 

where  by  §  334  the  quantity  in  square  brackets  is  the  square 
of  the  distance  8  from  the  line  P1P2  to  the  origin  0  (Fig.  139). 
Our  condition  means  therefore  that  the  line  P1P2  meets  the 
sphere  in  two  different  points,  touches  it,  or  does  not  meet  it 
at  all  according  as 

which  is  obvious  geometrically. 


324  SOLID  ANALYTIC  GEOMETRY        [XV,  §  345 

345.  Tangent  Cone.  The  condition  for  the  line  P^P^  to  be 
tangent  to  the  sphere  is  (§  344)  : 

W+  yi'+z,'-r')l(x,-x,y  +  (2/2  -  ^i)^  +  (^2  -  ^i)']. 
To  give  this  expression  a  more  symmetric  form  let  us  put,  to 
abbreviate, 

X1X2  +  2/1^2  +  2;i2;2  =  p,      a^i'  +  2/1'  +  ^i"  =  gu      ^2'  +  2/2'  4-  ^2'  =  92, 
so  that  the  condition  is 

(p-qiY.  =  (qi-r'){q,-2p  +  q2)y 
i.e.  p^  —  2  r^p  =  q^q^  —  r'^q^  —  r^^g  5 

adding  r*  in  both  members,  we  have 

i.e. 

{x^X2  +  2/12/2  +  z,z^  -  r')'  =  (a?i'  -f  2/1'  +  =2i'  -  r'){x^^  +  2/2'  +  ^2'  -  r"). 

Now  keeping  the  sphere  and  the  point  Pj  fixed,  let  Pg  vary 
subject  only  to  this  condition,  i.e.  to  the 
condition  that  P^P^  shall  be  tangent  to 
the  sphere;  the  point  Pg,  which  we  shall 
now  call  P{x^  y,  z)  is  then  any  point  of 
the  cone  of  vertex  Pj  tangent  to  the  cone.  ?i 
Hence   the   equation  of  the  cone  of  vertex  Fia.  139 

-f*i(^i  f  2/1 J  ^1)  tangent  to  the  sphere  x^  +  y^ -\- z'^  =  r^  is 

i^i'  +  2/1'  +  ^1'  -  r'Xx'  +  7f-\-z'-r')  =  {xix  +  2/12/  +  z,z  -  r^f. 

If,  in  particular,  the  point  Pi  is  taken  on  the  sphere  so  that 
^\  +  yi  +  z-^  =  r^,  the  equation  of  the  tangent  cone  reduces  to 
the  form  ^,^  +  y,y  +  ,,,  =  ^, 

which  represents  the  tangent  plane  at  P^. 

346.  Inversion.  A  sphere  of  center  0  and  radius  a  being  given, 
we  can  find  to  every  point  P  of  space  (excepting  0)  one  and  only  one 


XV,  §  346]  THE  SPHERE  325 

point  P'  on  OP  (produced  if  necessary)  such  that 

OP'  OP'  =  aK 
The  points  P,  P'  are  said  to  be  inverse  to  each  other  with  respect  to  the 
sphere  (compare  §  91). 

Taking  rectangular  axes  through  0,  we  find  as  the  relations  between 
the  coordinates  of  the  two  inverse  points  P(x,  y,  z)  and  P'{x\  y',  z')  if 
we  put  OP  =  r  =  Va:2  +  y^  +  z^.   OP'  =  r'  =  v'x'=^  +  y'^  +  z'^  : 

x_y'  _z'  _r'  _  rr'  _  a^  . 
X      y      z      r       f^      r'^  ' 

hence     x'  -         ^"^  y'-—^ z' - ^ • 

hence    ^-^2  +  ^2  +  ^2'         ^"^2  +  ^2  +  ^2'        ^-:,2  +  ^2  +  ^2' 

and  similarly 


y  = 


x'-^ -^  y'^ -}- z'^  x'-2  +  y'-^  +  z'-^  x''^  +  y'^  +  z'-^ 

These  equations  enable  us  to  find  to  any  surface  whose  equation  is  given 
the  equation  of  the  inverse  surface,  by  simply  substituting  for  x,  y,  z 
their  values. 

Thus  it  can  be  shown,  that  by  inversion  every  sphere  is  transformed 
into  a  sphere  or  a  plane.  The  proof  is  similar  to  the  corresponding  propo- 
sition in  plane  analytic  geometry  (§  92)  and  is  left  as  an  exercise. 

EXERCISES 

1.  Find  the  radius  of  the  circle  which  is  the  intersection :  (a)  of  the 
plane  y  =  Q  with  the  sphere  x^  +  y^  -\-  z^  —  6y  =  0  ;  (6)  of  the  plane 
2x—Sy  +  z-2  =  0  with  the  spherfe  x^  +  y'^  +  z^  -6x +  2y  -  lb  =  0. 

2.  A  line  perpendicular  to  the  plane  of  a  circle  through  its  center  is 
called  the  axis  of  the  circle.  Find  the  circle  :  (a)  which  lies  in  the  plane 
z  =  4:,  has  a  radius  3  and  Oz  as  axis ;  (6)  which  lies  in  the  plane  2/  =  5, 
has  a  radius  2  and  the  line  x  —  3  =  0,  0  —  4=0  as  axis. 

3.  Find  the  circles  of  radius  3  on  the  sphere  of  radius  4  about  the 
origin  whose  common  axis  is  equally  inclined  to  the  coordinate  axes. 

4.  Does  the  line  joining  the  points  (2,  —  1,  —  6),  (-  1,  2,  3)  intersect 
the  sphere  x^  +  y'^  +  z^  =  10?     Find  the  points  of  intersection. 


326  SOLID  ANALYTIC  GEOMETRY        [XV,  §  346 

5.  Find  the  planes  tangent  to  the  following  spheres  at  the  given 
points  :     (a)  x'^ +  y^ -{- z'^ -Sy  -  5z  ~2  =  0,  at    (2,  -  1,  3)  ; 

(6)  a;2  +  2/2  4. 2.2  _|_  2  X  -  6  y  +  2!  -1  =  0,  at  (0,  1,  -  3)  ; 

(c)  S{x^-\-y^  +  z^)-5x  +  2y  -  z  =  0,  at  the  origin  ; 

(d)  a;2  +  y2  ^z'^^ax-  by  -cz  =  0,  at  (a,  6,  c). 

6.  Find  the  tangent  cone  :  (a)  from  (4,  1,  —  2)  to  x^  -\- y^ -\-  z^  =  Q  ; 
(&)  from  (2  a,  0,  0)  to  x^  +  y^  +  z^  =  a^]  (c)  from  (4,  4,  4)  to  x^  +  if 
+  «2  _  16 ;  (e^)  from  (1,  -  5,  3)  to  x^  +  y^-\-z'^  =  9. 

7.  Find  the  cone  with  vertex  at  the  origin  tangent  to  the  sphere 
(x-2ay-\-y^  +  z^  =  a^. 

8.  Show  that,  by  inversion  with  respect  to  the  sphere  x^  -\-  y'^  +  ^2  _  ^52^ 
every  plane  (except  one  through  the  center)  is  transformed  into  a  sphere 
passing  through  the  origin. 

9.  With  respect  to  the  sphere  x"^ -}■  y^  +  z^  =  25,  find  the  surfaces  in- 
verse to  (a)  x  =  6,  (6)  x-y  =  0,  (c)  4  (x^  +  y^  -\- z^)-20  x-25  =  0. 

10.  Show  that  by  inversion  with  respect  to  the  sphere  ^2  -]- y^ -}■  z^  =  cfi 
every  line  through  the  origin  is  transformed  into  itself. 

11.  With  respect  to  the  sphere  x'^  -\-y'^  -\-  z^  =  a^,  find  the  surface  in- 
verse to  the  plane  tangent  at  the  point  Pi  (xi ,  yi ,  Zi). 

12.  Show  that  all  spheres  with  center  at  the  center  of  inversion  are 
transformed  into  concentric  spheres  by  inversion. 

13.  What  is  the  curve  inverse  to  the  circle  a;2  -f  y2  _|_  ^2  _  25,  0  =  4, 
with  respect  to  the  sphere  a;2  +  ^2  _|_  ^2  _  iq  9 

347.  Poles  and  Polars.  Let  P  and  P>  be  inverse  points  with 
respect  to  a  given  sphere  ;  then  the  plane  tt  through  P',  at  right  angles  to 
OP  ( 0  being  the  center  of  the  sphere) ,  is  called  the  polar  plane  of  the 
point  P,  and  P  is  called  the  pole  of  the  plane  tt,  with  respect  to  the 
sphere. 

With  respect  to  a  sphere  of  radius  a,  with  center  at  the  origin^ 

ic2  +  2/2  +  2^2  =  a^, 

the  equation  of  the  polar  plane  of  any  point  P\  {x\ ,  y^  Z\)   is  readily 
found  by  observing  that  its  distance  from  the  origin  is  a2/ri,  and  that  the 


XV,  §  349]  THE  SPHERE  •  327 

direction  cosines  of  its  normal  are  equal  to  xi/n,  yi/n,  z\fr\<^  where 
r^  =  xi'^^-  y-^  +  Z'^  ;  the  equation  is  therefore 
x\x  +  y\y  +  z\.z  =  a2. 
If,  in  particular,  the  point  Pi  lies  on  the  sphere,  this  equation,  by  §  343 
(5),  represents  the  tangent  plane  at  Pi.  Hence  the  polar  plane  of  any 
point  of  the  sphere  is  the  tangent  plane  at  that  point ;  this  also  follows 
from  the  definition  of  the  polar  plane. 

348.  With  respect  to  the  same  sphere  the  polar  planes  of  any  two 
points  Pi(a;i ,  yi ,  zi)  and  P2(iC2 ,  1/2 ,  Z2)  are 

xix  +  yiy  +  ziz  =  a2  and  X2X  +  y2y  +  z^z  =  a^. 

Now  the  condition  for  the  polar  plane  of  Pi  to  pass  through  P2  is 
a^ia;2  +  ym  +  Z1Z2  =  <jfi  ; 
but  this  is  also  the  condition  for  the  polar  plane  of  P2  to  pass  through  Pi. 
Hence  the  polar  planes  of  all  the  points  of  any  plane  w  (not  passing 
through  the  origin  0)  pass  through  a  common  point,  namely,  the  pole 
of  the  plane  ir ;  and  conversely,  the  poles  of  all  the  planes  through  a  com- 
mon point  P  lie  in  a  plane,  namely,  the  polar  plane  of  P. 

349.  The  polar  plane  of  any  point  P  of  the  line  determined  by  two 
given  points  Pi(xi ,  yi ,  zi)  and  P2(a;2 ,  t/2 ,  Z2)  (always  with  respect  to  the 
same  sphere  x^  -{■  y^  +  z^  =  a^)  is 

Ixi  +  k(X2  -xi)]x-{-  [yi  +  k(^y2  -  yi)]y  +  [zi+  k{z2  -  zi)'[z  =  a\ 

This  equation  can  be  written  in  the  form 

k. 

xix  +  yxy  +  ziz  —  a'^  +  - — -  {X2X  +  y2y  +  Z2Z  —  a^)  =  0, 

1  —  fC 

which  for  a  variable  k  represents  the  planes  of  the  pencil  whose  axis  is  the 
intersection  of  the  polar  planes  of  Pi  and  P2.  Hence  the  polar  planes  of 
all  the  points  of  a  line  X  pass  through  a  common  line ;  and  conversely, 
the  poles  of  all  the  planes  of  a  pencil  lie  on  a  line. 

Two  lines  related  in  this  way  are  called  conjugate  lines  (or  conjugate 
axes,  reciprocal  polars).     Thus  the  line  P1P2 

x-x\  _  y  —yx  _  z  -  z\ 


X2  -  Xi        2/2  —  yx        Z2  —  Zx 


328  SOLID   ANALYTIC  GEOMETRY        [XV,  §  349 

and  the  line  xix  +  yiy  +  ziz  =  a'^, 

X2X  +  ViV  +  ZiZ  =  a^ 
are  conjugate  with  respect  to  the  sphere  x^  +  y2  ^  ^2  _  ^^2, 
As  the  direction  cosines  of  these  lines  are  proportional  to 
X2  —  X1,   yi-  y\,   ^2  -  zi 


and 


yi    zi 
2/2    ^2 


\x\    y\\ 
I  Xi    yi  \ 


Z\      X\ 
Z2      Xl 

respectively,  the  two  conjugate  lines  are  at  right  angles  (§  331). 

350.  By  the  method  used  in  the  corresponding  problem  in  the  plane 
(§  95)  it  can  be  shown  that  the  polar  plane  of  any  point  P\{x\ ,  y\ ,  z\) 
with  respect  to  any  sphere 

^(^2  +  1/2  _|_  2;2)  +  2  G^X  +  2  ^2/  +  2  /^  +  ^"1=  0 

is 

A{xxx  +  yxy  +  zxz)  +  G{xx  +  x)  +  H{yx  +  y)  +  I{zx  +  0)  +  jr  =  0. 

351.  Power  of  a  Point,     if  in  the  left-hand  member  of  the  equation 

of  the  sphere 

(X  -  Kf  +  (2/  -  j)2  +  (^  -  kY  -  r2  =:  0 

we  substitute  for  x,  ?/,  0,  the  coordinates  xi ,  yi ,  ^1  of  any  point  not  on 
the  sphere,  we  obtain  an  expression  (xi  —  uy  +  (yi  —  j)2+  (s^i  —  A;)2  —  r2 
different  from  zero  which  is  called  the  power  of  the  point  Pi (xi  ^  yi,  zi^ 
with  respect  to  the  sphere. 

As  (xi  —  7i)2  +  (yi  —  j)2  +  (zi  -  A:)2  is  the  square  of  the  distance  d  be- 
tween the  point  Pi  and  the  center  C  of  the  sphere,  we  can  write  the 

power  of  Pi  briefly 

<Z2  -  r2 ; 

the  power  of  Pi   is  positive  or  negative  according  as  Pi  lies  outside  or 

within  the  sphere.      For  a  point  Pi  outside,  the  power  is  evidently  the 

square  of  the  length  of  a  tangent  drawn  from  Pi  to  the  sphere. 

352.  Radical  Plane,  Axis,  Center.  The  locus  of  a  point  whose 
powers  with  respect  to  the  two  spheres 

a:2  +  ?/2  +  22  +  a^x  +  biy  +  ciz  +  cZi  =  0, 
x^+y^  +  z^  +  aix  +  biy  +  C2Z  ^  d2  =  Q 
are  equal  is  evidently  the  plane 

(ai  —  a2)x  +  (5i  —  h2)y  +  (ci  -  C2)z  +  tZi  —  ^2  =  0, 
which  is  called  the  radical  plane  of  the  two  spheres.     It  always  exists  un- 
less the  two  spheres  are  concentric. 


XV,  §353]  THE  SPHERE  329 

It  is  easily  proved  that  the  three  radical  planes  of  any  three  spheres 
(no  two  of  which  are  concentric)  are  planes  of  the  same  pencil  (§  323)  ; 
and  hence  that  the  locus  of  the  points  of  equal  power  with  respect  to 
three  spheres  is  a  straight  line.  This  line  is  called  the  radical  axis  of  the 
three  spheres  ;  it  exists  unless  the  centers  lie  in  a  straight  line. 

The  six  radical  planes  of  four  spheres,  taken  in  pairs,  are  in  general 
planes  of  a  sheaf  (§  324) .  Hence  there  is  in  general  but  one  point  of 
equal  power  with  respect  to  four  spheres.  This  point,  the  radical  center 
of  the  four  spheres,  exists  unless  the  f our'centers  lie  in  a  plane. 

353.  Family  of  Spheres.    The  equation 

represents  a  family^  or  pencil,  of  spheres^  provided  k  ^—1.     If  the  two 

spheres 

x2  +  2/2  +  z^  +  «ix  +  hiy  +  ciz  +  (^i  =  0, 

X2  +  ?/2  +  2r2  +  a2.X  +  b^y  +  C2^  -h  ^2  =  0 

intersect,  every  sphere  of  the  pencil  passes  through  the  common  circle  of 
these  two  spheres.  If  ^•  =  —  1,  the  equation  represents  the  radical  plane 
of  the  two  spheres. 

EXERCISES 

1.  Find  the  radius  of  the  circle  in  which  the  polar  plane  of  the  point 
(4,  3,  —  1)  with  respect  to  x:^-\-y'^-\-z^  =  16  cuts  the  sphere. 

2.  Find  the  radius  of  the  circle  in  which  the  polar  plane  of  the  point 
(5,  —  1,  2)  with  respect  to  x'^  -{■  y'^  +  z"^  —  2x  +  ^y  =  ^  cuts  the  sphere. 

3.  Show  that  the  plane  3ic  +  ?/— 4s  =  19  is  tangent  to  the  sphere 
x'^  +  y^  +  z'^  —  2x  —  ^y  —  Qz—  \2,=0^  and  find  the  point  of  contact. 

4.  If  a  point  describes  the  plane  4  x  —  5  ?/  —  3  a:  =  16,  find  the  coordi- 
nates of  that  point  about  which  the  polar  plane  of  the  point  turns  with 
respect  to  the  sphere  aj2  -f  y2  ^  ^2  =  16. 

5.  If  a  point  describes  the  plane  2a:  +  3y  +  5!  =  4,  find  that  point 
about  which  the  polar  plane  of  the  point  turns  with  respect  to  the  sphere 
x2  +  y2  _|_  2-2  _  8. 

6.  If  a  point  describes  the  line  ^  ~     =  ^-i—  =  ^  ~     ,  find  the  equa- 

o  5  —  2i 

tions  of  that  line  about  which,  the  polar  plane  of  the  point  turns  with 


330  SOLID  ANALYTIC  GEOMETRY        [XV,  §  353 

respect  to  the  sphere  x^  +  y^  +  z^  =  25.      Show  that  the  two  lines  are 
perpendicular. 

7.  If  a  point  describe  the  line  2x-Sy-\-iz  =  2,  x  +  y  -{-  z  =  S,  find 
the  equations  of  that  line  about  which  the  polar  plane  of  the  point  turns 
with  respect  to  the  sphere  x"^  +  y^  -{-  z'^  =  16.  Show  that  the  two  lines  are 
perpendicular. 

8.  Find  the  sphere  through  the  origin  that  passes  through  the  circle 
of  intersection  of  the  spheres  x^  -\-  y'^+z"^  —  3  x  -^  i  y  —  6  z  —  8  =  0,  x^-\-y'^ 
-{-  z^  -  2  X  +  y  ~  z  -  10  =  0. 

9.  Show  that  the  locus  of  a  point  whose  powers  with  respect  to  two 
given  spheres  have  a  constant  ratio  is  a  sphere  except  when  the  ratio  is 
unity. 

10.  Show  that  the  radical  plane  of  two  spheres  is  perpendicular  to  the 
line  joining  their  centers. 

11.  Show  that  the  radical  plane  of  two  spheres  tangent  internally  or 
externally  is  their  common  tangent  plane. 

12.  Find  the  equations  of  the  radical  axis  of  the  spheres  x^  -\-  y^+  z^ 
-Sx-2y  -z-^  =  0,  x'^+y^  +  z^+5x-Sy-2z-S  =  0,  x^  +  ^a 
+  02  _  16  =  0. 

13.  Find  the  radical  ^center  of  the  spheres  x^  -\-y'^  +  z^  —  6  x  -{- 2y 

-  ;2  +  6  =  O;  5C2  +  ?/2  +  02  _  10  =  0,  X2  +  1/2  +  02  +  2  x  -  3  y  +  5  2!  -  6  =  0, 

x^  +  y^  -{-  z^  -  2x  +  4 y  -  12  =0. 

14.  Show  that  the  three  radical  planes  of  three  spheres  are  planes  of 
the  same  pencil. 

15.  Two  spheres  are  said  to  be  orthogonal  when  their  tangent  planes 
at  every  point  of  their  circle  of  intersection  are  perpendicular.  Show 
that  the  two  spheres  x^  -}-y^-^  z^  +  aix  +  biy  +  Ciz  +  t^i  =  0,  x2  -\-  y"^  +  z'^ 
+  a^x  +  hiy  +  C20  +  0^2  =  0  are  orthogonal  when  aia^  +  6160  +  C1C2 
=  2(di  +  (?2). 

16.  Write  the  equation  of  the  cone  tangent  to  the  sphere  x^  +  y'^  ■\- 
-5.2  — 1.2  ^itii  vertex  (0,  0,  z\).  Divide  this  equation  by  zi^  and  let  the 
vertex  recede  indefinitely,  i.e.  let  01  increase  indefinitely.  The  equation 
a;2  4-  2/2  =  |.2^  thus  obtained,  represents  the  cylinder  with  axis  along  the 
axis  Oz  and  tangent  to  the  sphere  x^  4-  y'^  -\-  z^  =  r'^. 


XV,  §  353]  THE  SPHERE  331 

17.  In  the  equation  of  the  tangent  cone  (§  345)  write  for  the 
coordinates  of  the  vertex  xi  =  nh ,  l/i  =  nnii ,  zi  =  rini ;  divide  the  equa- 
tion by  n^  and  let  n  increase  indefinitely,  i.e.  let  the  vertex  of  the  cone 
recede  indefinitely.  The  tangent  cone  thus  becomes  a  tangent  cylinder 
with  axis  passing  through  the  center  of  the  sphere  and  having  the  direc- 
tion cosines  h,  mi,  ni.     Show  that  this  tangent  cylinder  is 

(hx  +  miy  +  nizy^  -  {x^  +  y^  +  z^  -  r^)  =  0. 

18.  From  the  result  of  Ex.  17,  find  the  cylinder  with  axis  equally 
inclined  to  the  coordinate  axes  which  is  tangent  to  the  sphere  x^  -\-  y^ 
-\-z^  =  r2. 

19.  From  the  result  of  Ex.  17,  find  the  cylinders  with  axes  along  the 
coordinate  axes  which  are  tangent  to  the  sphere  x'^  -{-y"^  +  z^  =  r^. 

20.  Find  the  cylinder  with  axis  through  the  origin  which  is  tangent  to 
the  sphere  x^ -{-y'^  +  z'^  —  4:X  +  6y  —  S  =  0. 

21.  Find  the  family  of  spheres  inscribed  in  the  cylinder 

{Ix  +  my-\-  nzy  -  (x?-  -\-y^-\-z'^-  r^)  =  0. 

22.  Find  the  cylinder  with  axis  having  direction  cosines  Z,  m,  n  which 
is  tangent  to  the  sphere  (x  —  h)'^  +(y  —  j)'^  -\-{z  —  k)^  =  r^. 

23.  Show  that  as  the  point  P  recedes  indefinitely  from  the  origin  along 
a  line  through  the  origin  of  direction  cosines  Z,  m,  n,  the  polar  plane  of  P 
with  respect  to  the  sphere  x^  -f  y'^  +  z^  =  a^  becomes  ultimately  Ix  +  my 
+  nz  =  0. 


CHAPTER   XVI 
QUADRIC   SURFACES 

354.   The    Ellipsoid.      The    surface    represented    by    the 
equation 

is  called  an  ellipsoid.     Its  shape  is  best  investigated  by  tak- 
ing cross-sections  at  right  angles  to  the  axes  of  coordinates. 

Thus  the  coordinate  plane  Oyz  whose  equation  is  ic  =  0  in- 
tersects the  ellipsoid  in  the  ellipse 


Any  other  plane  perpendicular  to  the  axis  Ox  (Fig.  140),  at 

y 


Fig.  140 

the  distance  h  ^  a  from  the  plane  Oyz  intersects  the  ellipsoid 
in  an  ellipse  whose  equation  is 

7i' 


^2  +  ^2--^       .2' 


I.e. 


f 


K'-S)  -(•--:) 


=  1. 


332 


XVI,  §  355]  QUADRIC  SURFACES  333 

Strictly  speaking  this  is  the  equation  of  the  cylinder  that  pro- 
jects the  cross-section  on  the  plane  Oyz.  But  it  can  also  be 
interpreted  as  the  equation  of  the  cross-section  itself,  referred 
to  the  point  Qi,  0,  0)  as  origin  and  axes  in  the  cross-section 
parallel  to  Oy  and  Oz. 

Notice  that  as  h  <  a,  Jv^/a^,  and  hence  also  1  —  h^/a"^,  is  a  posi- 
tive proper  fraction.  The  semi-axes  6Vl  —  h^/a^,  c VI  —  h'^/a'^ 
of  the  cross-section  are  therefore  less  than  b  and  c,  respec- 
tively. As  h  increases  from  0  to  a,  these  semi-axes  gradually 
diminish  from  b,  c  to  0. 

355.  Cross-Sections.  Cross-sections  on  the  opposite  side 
of  the  plane  Oyz  give  the  same  results ;  the  ellipsoid  is  evi- 
dently symmetric  with  respect  to  the  plane  Oyz. 

By  the  same  method  we  find  that  cross-sections  perpendicu- 
lar to  the  axes  Oy  and  Oz  give  ellipses  with  semi-axes  dimin- 
ishing as  we  recede  from  the  origin.  The  surface  is  evidently 
symmetric  to  each  of  the  coordinate  planes.  It  follows  that 
the  origin  is  a  center,  i.e.  every  chord  through  that  point  is 
bisected  at  that  point.  In  other  words,  if  (x,  y,  z)  is  a  point 
of  the  surface,  so  is  {—x,  —y,  —z).  Indeed,  it  is  clear  from 
the  equation  that  if  {x,  y,  z)  lies  on  the  ellipsoid,  so  do  the 
seven  other  points  {x,  y,  -2),  {x,  -y,  z),  {-x,  y,  z),  {x,  -y,  -z), 
(—x,y,  —z),  (—X,  —y,  z),  {—x,—y,—z).  A  chord  through 
the  center  is  called  a  diameter. 

It  follows  that  it  suffices  to  study  the  shape  of  the  portion  of 
the  surface  contained  in  one  octant,  say  that  contained  in  the  tri- 
hedral formed  by  the  positive  axes  Ox,  Oy,  Oz ;  the  remaining 
portions  are  then  obtained  by  reflection  in  the  coordinate  planes. 

The  ellipsoid  is  a  dosed  surface;  it  does  not  extend  to  in- 
finity ;  indeed  it  is  completely  contained  within  the  parallel- 
epiped with  center  at  the  origin  and  edges  2  a,  2  &,  2  c,  parallel 
to  Ox,  Oy,  Oz,  respectively. 


334 


SOLID  ANALYTIC  GEOMETRY      [XVI,  §  356 


356.  Special  Cases.  In  general,  the  semi-axes  a,  h,  c  of  the 
ellipsoid,  i.e.  the  intercepts  made  by  it  on  the  axes  of  coordi- 
nates, are  different.  But  it  may  happen  that  two  of  them,  or 
even  all  three,  are  equal. 

In  the  latter  case,  i.e.  if  a  =  h  =  c,  the  ellipsoid  evidently 
reduces  to  a  sphere. 

If  two  of  the  axes  are  equal,  e.g.  if  &  =  c,  the  surface 

a"      ¥      b^ 


Fig.  141 


is  called  an  ellipsoid  of  revolution  because  it  can  be  generated 
by  revolving  the  ellipse 

y\ 

about  the  axis    Ox  (Fig.   141). 

Any  cross-section  at  right  angles 

to  Ox,  the  axis  of  revolution,  is  a   ^ 

circle,  while  the  cross-sections  at 

right  angles  to   Oy  and   Oz  are 

ellipses.     The  circular  cross-section  in  the  plane  Oyz  is  called 

the  equator;  the  intersections  of  the  surface  with  the  axis  of 

revolution  are  the  poles. 

If  a  >  6  (a  being  the  intercept  on  the  axis  of  revolution), 
the  ellipsoid  of  revolution  is  called  prolate;  if  a  <  b,  it  is 
called  oblate.  In  astronomy  the  ellipsoid  of  revolution  is 
often  called  spheroid,  the  surfaces  of  the  planets  which  are 
approximately  ellipsoids  of  revolution  being  nearly  spherical. 
Thus  for  the  surface  of  the  earth  the  major  semi-axis,  i.e.  the 
radius  of  the  equator,  is  3962.8  miles  while  the  minor  semi- 
axis,  i.e.  the  distance  from  the  center  to  the  north  or  south 
pole,  is  3949.6  miles. 


XVI,  §  357] 


QUADRIC  SURFACES 


335 


367.  Surfaces  of  Revolution.  A  surface  that  can  be  gen- 
erated by  the  revolution  of  a  plane  curve  about  a  line  in  the 
plane  of  the  curve  is  called  a  surface  of  revolution.  Any  such 
surface  is  fully  determined  by  the  generating  curve  and  the 
position  of  the  axis  of  revolution  with  respect  to  the  curve. 

Let  us  take  the  axis  of  revolution  as  axis  Ox,  and  let  the 
equation  of  the  generating  curve  be 

As  this  curve  revolves  about  Ox,  any 
point  P  of  the  curve  (Fig.  142)  de- 
scribes a  circle  about  Ox  as  axis, 
with  a  radius  equal  to  the  ordinate 
f{x)  of  the  generating  curve.  For 
any  position  of  P  we  have  therefore 

and  this  is  the  equation  of  the  surface  of  revolution. 
Thus  if  the  ellipse 

a^     h-' 
revolves  about  the  axis  Ox,  we  find  since  y  =  ±  (b/a)  Va' 
for  the  ellipsoid  of  revolution  so  generated  the  equation 


Fig.  142 


a;2 


a 


x^), 


which  agrees  with  that  of  §  356. 

Any  section  of  a  surface  of  revolution  at  right  angles  to  the 
axis  of  revolution  is  of  course  a  circle ;  these  sections  are  called 
parallel  circles,  or  simply  parallels  (as  on  the  earth's  surface). 
Any  section  of  a  surface  of  revolution  by  a  plane  passing 
through  the  axis  of  revolution  is  called  a  meridian  section  ; 
it  consists  of  the  generating  curve  and  its  reflection  in  the  axis 
of  revolution. 


336  SOLID  ANALYTIC  GEOMETRY      [XVI,  §  357 

EXERCISES 

—  1.  An  ellipsoid  has  six /oci,  viz.  the  foci  of  the  three  ellipses  in  which 
the  ellipsoid  is  intersected  by  its  planes  of  symmetry.  Determine  the 
coordinates  of  these  foci :  (a)  for  an  ellipsoid  with  semi-axes  1,  2,  3 ; 
(6)  for  the  earth  (see  §356)  ;  (c)  for  an  ellipsoid  of  semi-axes  10,  8,  1 ; 
(d)  for  an  ellipsoid  of  semi-axes  1,  1,  5. 

2.  Show  that  the  intersection  of  an  ellipsoid  with  any  plane  actually 
cutting  the  ellipsoid  is  an  ellipse  by  proving  that  the  projection  of  this 
curve  of  intersection  on  each  coordinate  plane  is  an  eUipse. 

3.  Assuming  a  >  &  >  c  in  the  equation  of  §  354  find  the  planes  through 
Oy  that  mtersect  the  ellipsoid  in  circles. 

'-  "  4.  Find  the  equation  of  the  paraboloid  of  revolution  generated  by  the 
revolution  of  the  parabola  y'^  =  4:  ax  about  Ox. 

6.  Find  the  equation  of  a  torus,  or  anchor-ring,  i.e.  the  surface 
generated  by  the  revolution  of  a  circle  of  radius  a  about  a  line  in  its  plane 
at  the  distance  b>  a  from  its  center. 

6.    Find  the  equation  of  the  surface  generated  by  the  revolution  of  a 
.  circle  of  radius  a  about  a  line  in  its  plane  at  the  distance  &  <  a  from  its 
center.     Is  the  appearance  of  this  surface  noticeably  different  from  the 
surface  of  Ex.  5  ? 

7.   Show  what  happens  to  the  surface  of  Ex.  6  when  6  =  0;  when  &  =  a. 

8.  Find  the  equation  of  the  surface  generated  by  the  revolution  of  the 
parabola  y^  =  4tax  about:  (a)  the  tangent  at  the  vertex;  (&)  the  latus 
rectum. 

"*  9.  Find  the  equation  of  the  surface  generated  by  the  revolution  of  the 
hyperbola  xy  =  a^  about  an  asymptote. 

10.  Find  the  cone  generated  by  the  revolution  of  the  line  y  =  mx  -{-  b 
about:  (a)   Ox,  (6)   Oy. 

11.  How  are  the  following  surfaces  of  revolution  generated  ? 

(a)  y^+z^=x^.  (&)  2x^+2y^-^z=0.  (c)  x^-\-y^-z'^-2x+i  =  0. 

12.  Find  the  equation  of  the  surface  generated  by  the  revolution  of 
the  ellipse  x^  +  4  ?/2  —  4  x  =  0  :  (a)  about  the  major  axis  ;  (b)  about  the 
minor  axis  ;  (c)  about  the  tangent  at  the  origin. 


XVI,  §  359] 


QUADRIC  SURFACES 


337 


358.   Hyperboloid  of  One  Sheet.     The  surface  represented 
by  the  equation 


=  1 


a"     y^     & 
is  called  a  hyperboloid  of  one  sheet  (Fig.  143).    The  intercepts 


Fig.  143 

on  the  axes  Ox,  Oy  are  ±a,  ±  6 ;  the  axis  Oz  does  not  intersect 
the  surface. 

359.   Cross-Sections.     The  plane  Oxy  intersects  the  surface 
in  the  ellipse 

cross-sections  perpendicular  to  Oz  give  ellipses  with  ever-in- 
creasing semi-axes. 
The  planes  Oyz  and  Ozx  intersect  the  surface  in  the  hyperbolas 

Any  plane  perpendicular  to  Ox,  at  the  distance  h  from   the 
origin,  intersects  the  hyperboloid  in  a  hyperbola,  viz. 

f  ■     z' 


338  SOLID  ANALYTIC  GEOMETRY      [XYI,  §  359 

As  long  as  /i  <  a  this  hyperbola  has  its  transverse  axis  parallel 
to  Oy  while  for  h  >  a  the  transverse  axis  is  parallel  to  Oz ;  for 
h  =  a  the  equation  reduces  to  y'^/h'^  —  z^/c^  =  0  and  represents 
two  straight  lines,  viz.  the  parallels  through  (a,  0,  0)  to  the 
asymptotes  of  the  hyperbola  y^/b"^  —  z^/x^  =  1  which  is  the 
intersection  of  the  surface  with  the  plane  Oyz. 

Similar  considerations  apply  to  the  cross-sections  perpen- 
dicular to  Oy. 

The  hyperboloid  has  the  same  properties  of  symmetry  as  the 
ellipsoid  (§  355)  ;  the  origin  is  a  center,  and  it  suffices  to  inves- 
tigate the  shape  of  the  surface  in  one  octant. 

360.  Hyperboloid  of  Revolution  of  one  Sheet.  If  in  the 
hyperboloid  of  one  sheet  we  have  a  =  b,  the  cross-sections  per- 
pendicular to  the  axis  Oz  are  all  circles  so  that  the  surface  can 
be  generated  by  the  revolution  of  the  hyperbola 

about  Oz.  Such  a  surface  is  called  a  hyperboloid  of  revolution 
of  one  sheet. 

361.  Other  Forms.     The  equations 

^-^'  +  -  =  1     -^  +  ^'  +  ?!  =  l 
a2     b^      d"       '        a^      b""      c^ 

also  represent  hyperboloids  of  one  sheet  which  can  be  investi- 
gated as  in  §§  358-360.     In  the  former  of  these  the  axis  Oy,  in 
the  latter  the  axis  Ox,  does  not  meet  the  surface. 
Every  hyperboloid  of  one  sheet  extends  to  infinity. 

362.  Hyperboloid  of  Two  Sheets.  The  surface  represented 
by  the  equation 

a"-     ¥     c2~ 
is  called  a  hyperboloid  of  two  sheets  (Fig.  144). 


XVI,  §  365] 


QUADRIC  SURFACES 


339 


The  intercepts  on  Ox  are  ±  a ;  the  axes  Oy,  Oz  do  not  meet 
the  surface. 

363.   Cross-Sections.     The  cross-sections  at  right  angles  to 
Ox,  at  the  distance  h  from  the  origin  are 

2/'  2;2  _ . 


(-S) 


C2fl 


these  are  imaginary  as  long  as  7i  <  a; 
for  h>a  they  are  ellipses  with  ever- 
increasing  semi-axes  as  we  recede  from 
the  origin. 

The  cross-sections  at  right  angles  to  Oy 
and  Oz  are  hyperbolas. 

The  hyperboloid  of  two  sheets,  like  that  of  one  sheet  and 
like  the  ellipsoid,  has  three  mutually  rectangular  planes  of 
symmetry  whose  intersection  is  therefore  a  center. 

The  surfaces 


Fig.  1M 


_^  _,]r     ^—i     ^^^yiA-^—i 


are  hyperboloids  of  two  sheets,  the  former  being  met  by  Oy, 
the  latter  by  Oz,  in  real  points. 

The  hyperboloid  of  two  sheets  extends  to  infinity. 

364.  Hyperboloid  of  Revolution  of  Two  Sheets.    If  &  =  c 

in  the  equation  of  §  362,  the  cross-sections  at  right  angles  to  Ox 
are  circles  and  the  surface  becomes  a  hyperboloid  of  revolution 
of  two  sheets.     . 

365.  Imaginary  Ellipsoid.     The  equation 

_x^  _y^  _z^  _A 

is  not  satisfied  by  any  point  with  real  coordinates.  It  is  some- 
times said  to  represent  an  imaginary  ellipsoid. 


340  SOLID  ANALYTIC  GEOMETRY      [XVI,  §  366 

366.  The  Paraboloids.    The  surfaces 


a;2      7/2_ 


a-      0^  a^     W 

which  are  called  the  elliptic  paraboloid  (Fig.  145)  and  hyper- 
bolic paraboloid  (Fig.  146),  respectively,  have  each  only  two 
planes  of  symmetry,  viz  the  planes  Oyz  and  Ozx.  We  here 
assume  that  c^O.       The  cross-sections  at  right  angles  to  the 


Fig.  145 


Fig.  146 


axis  Oz  are  evidently  ellipses  in  the  case  of  the  elliptic  parab- 
oloid, and  hyperbolas  in  the  case  of  the  hyperbolic  paraboloid. 
The  plane  Oxy  itself  has  only  the  origin  in  common  with  the 
elliptic  paraboloid ;  it  intersects  the  hyperbolic  paraboloid  in 
the  two  lines  x'^/a'^  —  y'^/b^  =  0,  i.e.  y  =  ±  bx/a. 

The  intersections  of  the  elliptic,  paraboloid  (Fig.  145)  with 
the  planes  Oyz  and  Ozx  are  parabolas  with  Oz  as  axis  and  0  as 
vertex,  opening  in  the  sense  of  positive  2  if  c  is  positive,  in  the 
sense  of  negative  2;  if  c  is  negative.  Planes  parallel  to  these 
coordinate  planes  intersect  the  elliptic  paraboloid  in  parabolas 
with  axes  parallel  to  Oz,  but  with  vertices  not  on  the  axes  Ox, 
Oy,  respectively. 

For  the  hyperbolifc  paraboloid  (Fig.  146),  which  is  saddle- 
shaped  at  the  origin,  the  intersections  with  the  planes  Oyz  and 


XVI,  §  369] 


QUADRIC  SURFACES 


341 


Ozx  are  also  parabolas  with  Oz  as  axis ;  if  c  is  positive  the 
parabola  in  the  plane  Oyz  opens  in  the  sense  of  negative  z,  that 
in  the  plane  Ozx  opens  in  the  sense  of  positive  z.  Similarly 
for  the  parallel  sections. 

367.  Paraboloid  of  Revolution.     If  in  the  equation  of  the 
elliptic  paraboloid  we  have  a=b,  it  reduces  to  the  form 

x^-\-y^  =  2pz. 

This  represents  a  surface  of  revolution,  called  the  paraboloid  of 
revolution.  This  surface  can  be  regarded  as  generated  by  the 
revolution  of  the  parabola  y'^  =  2pz  about  the  axis  Oz. 

368.  Elliptic  Cone.    The  surface  represented  by  the  equation 


x^     y^ 


=  0 


is  an  elliptic  cone,  with  the  origin  as  vertex  and  the  axis  Oz  as 
axis  (Fig.  147). 

The  plane  Oxy  has  only  the  origin  in 
common  with  the  surface.  Every  parallel 
plane  z  =  k,  whether  Jc  be  positive  or  negative, 
intersects  the  surface  in  an  ellipse,  with 
semi-axes  increasing  proportionally  to  k. 

The  plane  Oyz,  as  well  as  the  plane  Ozx, 
intersects  the  surface  in  two  straight  lines 
through  the  origin.  Every  plane  parallel  to 
Oyx  or  to  Ozx  intersects  the  surface  in  a 
hyperbola.  Fig.  147 

369.  Circular  Cone.  If  in  the  equation  of  the  elliptic  cone 
we  have  a  =  b,  the  cross-sections  at  right  angles  to  the  axis  Oz 
become  circles.     The  cone  is  then  an  ordinary  circular  cone,  or 


342  SOLID  ANALYTIC  GEOMETRY      [XVI,  §  369 

cone  of  revolution,  which  can  be  generated  by  the  revolution 
of  the  line  y  =  (^a/c)z  about  the  axis  Oz.  Putting  a/c  =  m  we 
can  write  the  equation  of  a  cone  of  revolution  about  Oz,  with 
vertex  at  0,  in  the  form 

370.  Quadric  Surfaces.  The  ellipsoid,  the  two  hyper- 
boloids,  the  two  paraboloids,  and  the  elliptic  cone  are  called 
quadric  surfaces  because  their  cartesian  equations  are  all  of 
the  second  degree. 

Let  us  now  try  to  determine,  conversely,  all  the  various  loci 
that  can  be  represented  by  the  general  equation  of  the  second 
degree 

Ax^  +  By^  +  (7^2  +  2  Dyz  +  2  Ezx  +  2  Fxy 

+  2  6x-{-2Hy  +  2Iz-{-J=0. 

In  studying  the  equation  of  the  second  degree  in  x  and  y 
(§  249)  it  was  shown  that  the  term  in  xy  can  always  be 
removed  by  turning  the  axes  about  the  origin  through  a  cer- 
tain angle.  Similarly,  it  can  be  shown  in  the  case  of  three 
variables  that  by  a  properly  selected  rotation  of  the  coordinate 
trihedral  about  the  origin  the  terms  in  yx,  zx,  xy  can  in  general 
all  be  removed  so  that  the  equation  reduces  to  the  form 

(1)         Ax^  +  Bi/^ +  Cz^ +2Gx  +  2Hy+2Iz-{-J-=0. 

•  This  transformation  being  somewhat  long  will  not  be  given 
here.  We  shall  proceed  to  classify  the  surfaces  represented 
by  equations  of  the  form  (1). 

371.  Classification.  The  equation  (1)  can  be  further  sim- 
plified by  completing  the  squares.  Three  cases  may  be  distin- 
guished according  as  the  coefficients  A,  B,  C  are  all  three  differ- 
ent from  zero,  one  only  is  zero,  or  two  are  zero. 


XVI,  §371]  QUADRIC  SURFACES  343 

Case  (a) :  ^  ^  0,  jB  ^  0,  C  ^0.  Completing  the  squares  in 
Xf  y,  z  we  find 

Referred  to  parallel  axes  through  the  point  (—  G/A,  —  H/B, 
—  I/C)  this  equation  becomes 

(2)  Ax''-\-Bf-]-Cz^  =  J,. 

Case  (6)  :  A=^0,B^O,  (7=0.  Completing  the  squares  in  x 
aud  y  we  find 

If  /^  0  we  can  transform  to  parallel  axes  through  the  point 
(—G/Aj  —  H/B,  J2/2 1)  so  that  the  equation  becomes 

(3)  Ax"  +  By^+2Iz  =  0. 

If,  however,  7=0,  we  obtain  by  transforming  to  the  point 
(-G/A,  -II/B,0) 

(3')  Aa^-\-By'=J,. 

Case  (c)  :  A^O,  B  =  Of  C  =  0.  Completing  the  square  in 
x  we  have 

A(x-{-^\2Hy  +  2Iz  =  ^-J=J,. 

If  H  and  I  are  not  both  zero  we  can  transform  to  parallel 
axes  through  the  point  (—  G/A,  J^/2  H,  0)  or  through  (—  G/A, 
0,  J3/2  /)  and  find 

(4)  Ax'-\-2Hy  +  2Iz  =  0. 

If  7r=  0  and  /=  0  we  transform  to  the  point  (—  G/A,  0,  0) 
so  that  we  find 

(4')  Ax'^J,, 


344  SOLID  ANALYTIC  GEOMETRY      [XVI,  §  372 

372.  Squared  Terms  all  Present.  Case  (a).  We  proceed  to 
discuss  the  loci  represented  by  (2).  If  J^  4^  0,  we  can  divide 
(2)  by  J^  and  obtain  : 

(a)  if  ^/t/i,  5/t7i,  (7/Ji  are  positive,  an  ellipsoid  (§  354) ; 

(fi)  if  two  of  these  coefficients  are  positive  while  the  third 
is  negative,  a  hyperboloid  of  one  sheet  (§  358) ; 

(y)  if  one  coefficient  is  positive  while  two  are  negative,  a 
hyperboloid  of  tivo  sheets  (§  362); 

(8)  if  all  three  coefficients  are  negative,  the  equation  is  not 
satisfied  by  any  real  point  (§  365) ; 

If  Ji  =  0  the  equation  (2)  represents  an  elliptic  cone  (§  368) 
unless  A,  B,  C  all  have  the  same  sign,  in  which  case  the  origin 
is  the  only  point  represented. 

373.  Case  (b).  The  equation  (3)  of  §371  evidently  fur- 
nishes the  two  paraboloids  (§  366)  ;  the  paraboloid  is  elliptic  if 
A  and  B  have  the  same  sign ;  it  is  hyperbolic  if  A  and  B  are  of 
opposite  sign. 

The  equation  (3')  since  it  does  not  contains  and  hence  leaves 
z  arbitrary  represents  the  cylinder ,  with  generators  parallel  to  Oz, 
passing  through  the  conic  Ax^  -f-  By"^  =  ./g.  As  A  and  B  are 
assumed  different  from  zero,  this  conic  is  an  ellipse  if  A/J2  and 
and  B/J2  are  both  positive,  a  hyperbola  if  A/J^  and  B/Jc,  are  of 
opposite  sign,  and  it  is  imaginary  if  A/J2  and  B/J<^  are  both 
negative.  This  assumes  Jg  ^  0.  If  J^  =  0,  the  conic  degen- 
erates into  two  straight  lines,  real  or  imaginary ;  the  cylinder 
degenerates  into  two  planes  if  the  lines  are  real. 

374.  Case  (c).  There  remain  equations  (4)  and  (4').  To  sim- 
plify (4)  we  may  turn  the  coordinate  trihedral  about  Ox  through 
an  angle  whose  tangent  is  —  H/I-,  this  is  done  by  putting 

Bf  +  Hz'  -  Hy'  +  Iz' 

x  =  x  \      y=    -^  z  —  —   ^  ; 

^H^+P  v'H^  +  P 


XVI,  §374]  QUADRIC  SURFACES  345 

our  equation  then  becomes 


It  evidently  represents  a  parabolic   cylinder,  with  generators 
parallel  to  Oy. 

Finally,  the  equation  (4')  is  readily  seen  to  represent  two 
planes  perpendicular  to  Ox,  real  or  imaginary,  unless  t/3  =  0 
in  which  case  it  represents  the  plane  Oyz. 

EXERCISES 

1.  Name  and  locate  the  following  surfaces  : 

{a)  x^-{-2y^  +  Sz^  =  4.  (b)  x^  +  y^  -  5z  -  6  =  0. 

(c)   x'^  -  y^ -\- z^  =  4.  (d)  x^-y^  +  z^  +  Sz  +  6  =  0. 

(e)    2?/2 -4^2  _  5=3  0.  (/)  2x2  +  2/2  +  3^2  +  5  =  0. 

(g)  6^2  +  2 x2  =  10.  (h)  z^-9  =  0. 

(0    x2  -  y  +  1  =  0.  (j)   x^-'y^-z^  +  Qz  =  9. 

(k)  x2  +  3  ?/2  +  2;2  +  4  0  +  4  =  0.  (I)    z'^  +  ?/;-  9  =  0. 

2.  The  cone 

x2/a-^  +  2/2/6-2  _  2-2/02  =  0 

is  called  the  asymptotic  cone  of  the  hyperboloid  of  one  sheet 

x2/a2  +  yl/yZ  _  22/c2  =  1. 

Show  that  as  z  increases  the  two  surfaces  approach  each  other,  i.e.  they 
bear  a  relation  similar  to  a  hyperbola  and  its  asymptotes. 

3.  What  is  the  asymptotic  cone  of  the  hyperboloid  of  two  sheets  ? 

4.  Show  that  the  intersection  of  a  hyperboloid  of  two  sheets  with  any 
plane  actually  cutting  the  surface  is  an  ellipse,  parabola,  or  hyperbola. 
Determine  the  position  of  the  plane  for  each  conic. 

5.  Show  that  in  general  nine  points  determine  a  quadric  surface  and 
that  the  equation  may  be  written  as  a  determinant  of  the  tenth  order 
equated  to  zero. 

6.  Show  that  the  surface  inverse  to  the  cylinder  x'^  ■\-  y"^  =  a"^^  with 
respect  to  the  sphere  ^2  +  ^/^  +  ^2  —  ^2^  jg  ^\^q  torus  generated  by  the  rev- 
olution of  the  circle  {y  —  a/2y^  -\-  z"^  =  a?-  about  the  axis  Ox. 

7.  Determine  the  nature  of  the  surface  xyz  =  a^  by  means  of  cross- 
sections. 


346  SOLID  ANALYTIC  GEOMETRY      [XVI,  §  375 

375.   Tangent  Plane  to  the  Ellipsoid.     The  plane  tangent 
to  the  ellipsoid 

a"     b^     c^ 

can  be  found  as  follows  (compare  §§  344,  345).  The  equa- 
tions of  the  line  joining  any  two  given  points  (x^,  y^,  z^)  and 
{^2,y2,^2)  are 

x  =  x^  +  'k{x^-x;)y  y  =  yi-\-k(y^-yi),  z  =  Zi-\-k(z2-Zi). 

This  line  will  be  tangent  to  the  ellipsoid  if  the  quadratic 
in  k 

o?  ¥  (? 

has  equal  roots.     Writing  this  quadratic  in  the  form 

1_       a^  b^  c2      J 

oFxiix^-x,)    yifa-yi)  I  gife-gQ"];,  I  W  I  yi"  .^i'    i\=() 
"^   [       a2        -^        52       ^       c2       J    ^\a''^b'^c\      J      ' 

we  find  the  condition 


[( 


"  52  ^2 


_r(x^  -  x,y    O/2  -  yiY  _,  fe  -  ZiYlf^i^  j_  ^  4.  5l  _  1 V 

\_      a"  6^       "^       c2      JVa2  "^  62  ^  c2        J 

If  now  we  keep  the  point  {Xi ,  2/1 , 2!i)  fixed,  but  let  the  point 
(X2,  yz,  Z2)  vary  subject  to  this  condition,  it  will  describe  the 
cone,  with  vertex  (x^ ,  2/1 ,  ^i),  tangent  to  the  ellipsoid ;  to  indi- 
cate this  we  shall  drop  the  subscripts  of  X2,  2/2,  Z2.  If,  in 
particular,  the  point  (xi ,  2/1 ,  z^)  be  chosen  on  the  ellipsoid,  we 
have 


XVI,  §377]  QUADRIC  SURFACES  347 

and  the  cone  becomes  the  tangent  plane.     The  equation  of  the 
tangent  plane  to  the  ellipsoid  at  the  point  {x^ ,  y^ ,  z^  is,  therefore  : 

a"       we' 

376.   Tangent  Planes  to  Hyperboloids.     In  the  same  way 
it  can  be  shown  that  the  tangent  planes  to  the  hyperboloids 


a""     h^     c2~    '    a2     y-     c2~ 
at  (aji ,  2/i ,  2i)  are 

a2  62  f.1  '       ^2  ^2  ^2 

By  an  equally  elementary,  but  somewhat  longer,  calculation 
it  can  be  shown  that  the  tangent  plane  to  the  quadric  surface 

Ax^  -\-  By^  -\-  Cz^  -{-2  Dyz  -{-2  Ezx-^2  Fxy 

-\-2Gx-\-2Hy  +  2Iz^J=0 
at  (xi ,  2/i ,  Zi)  is : 

AxiX  +  Byiy  +  Cz^z  +  D  {y^z  +  z^y)  +  E  (z^x  +  x^z)  -\-  F{x,y  +  ^/lO;) 
-{.G(x,-{-x)-^  H{y,  +  y)-\-  I(z,  +  z)  +  J=  0. 

In  particular,  the  tangent  planes  to  the  paraboloids 

t  +  t^2cz,    ''--t  =  2cz 
a"     b^  '    a2      52 

are 

^2  ^  52         ^  '  ^   ^'     a2        62         V  1  -r   ; 

377.  Ruled  Surfaces.  A  surface  that  can  be  generated  by 
the  motion  of  a  straight  line  is  called  a  ruled  surface;  the  line 
is  called  the  generator. 

The  plane  is  a  ruled  surface.  Among  the  quadric  surfaces 
not  only  the  cylinders  and  cones  but  also  the  hyperboloid  of 
one  sheet  and  the  hyperbolic  paraboloid  are  ruled  surfaces. 


348 


SOLID  ANALYTIC  GEOMETRY      [XVI,  §  378 


378.  Rulings  on  a  Hyperboloid  of  One  Sheet.    To  show 
this  for  the  hyperboloid 

a^     h"-     &       ' 
we  write  the  equation  in  the  form 

62         c2 


x^ 


and  factor  both  members  : 


»+' 


)e-9-^3('-3- 


It  is  then  apparent  that  any  point  whose  coordinates  satisfy 
the  two  equations 


^  +  5==;fcfl+^\    ^_^-l 


1 - 

/cV        a 


where  A;  is  an  arbitrary  parameter,  lies 
on  the  hyperboloid.  These  two  equa- 
tions represent  for  every  value  of  A:  (^  0) 
a  straight  line.  The  hyperboloid  of  one 
sheet  contains  therefore  the  family  of 
lines  represented  by  the  last  two  equa- 
tions with  variable  A;. 

In  exactly  the  same  way  it  is  shown  that  the  same  hyper- 
boloid also  contains  the  family  of  lines 

0      c        V        aj     0     c     k\       a  J 

Thus  every  hyperboloid  of  one  sheet  contains  two  sets  of  recti- 
linear generators  (Fig.  148). 


Fig.  148 


XVI,  §  379] 


QUADRIC   SURFACES 


349 


379.   Rulings  on  a  Hyperbolic  Paraboloid.    The  hyperbolic 
paraboloid  (Fig.  149) 


also  contains  tivo  sets  of  recti- 
linear generators,  namely, 

^  +  l  =  k.2cz,    2-1  =  1 
a      b  a     0     k 

and 
a     b  a     b     7c' 


Fig.  149 


EXERCISES 

1.  Derive  the  equation  of  the  tangent  plane  to  : 

(a)  the  elliptic  paraboloid  ;   (b)  the  hyperbolic  paraboloid ; 
(c)  the  elliptic  cone. 

2.  The  line  perpendicular  to  a  tangent  plane  at  a  point  of  contact  is 
called  the  normal  line.  Write  the  equations  of  the  tangent  planes  and 
normal  lines  to  the  following  quadric  surfaces  at  the  points  indicated : 

(a)  xy9  +  yyi  -  02/16  =  1,  at  (3,  -  1,  2)  ; 
(6)  x2  + 2  2/2 +  02  =  10,  at  (2,1,  -2); 
(c)  x2  +  2  ?/2  -  2  ^2  =  0,  at  (4,  1,  3) ;  (d)  x^-Sy^-z  =  0,  at  the  origin. 

3.  Show  that  the  cylinder  whose  axis  has  the  direction  cosines  I,  m,  n 
and  which  is  tangent  to  the  ellipsoid  od^a^  +  y^/b^  -f  z^/c^  =  1,  is 

W      b'^       cy        W     b^      d'JW     b'^     c^       I 


4.  Show  that  the  plane  Ix  +  my  -{-  nz  =  y/'V-a^  +  ?n2&2  +  ^12^2  jg  tangent 
to  the  ellipsoid  x'^la'^  +  2/2/52  +  ^ij^fi  _  1. 

5.  Show  that  the  locus  of  the  intersection  of  three  mutually  perpen- 
dicular tangent  planes  to  the  ellipsoid  x^ja'^  +  ?/2/62  _^  ^2/^2  =  1,  is  the 
sphere  (called  director  sphere)  x^  +  y^  +z^  =  a^  +  62  _|.  ^2. 


350  SOLID  ANALYTIC  GEOMETRY      [XVI,  §  379 

6.  Show  that  the  elliptic  cone  is  a  ruled  surface. 

7.  Show  that  any  two  linear  equations  which  contain  a  parameter 
represent  the  generating  line  of  a  ruled  surface.  What  surfaces  are  gen- 
erated by  the  following  lines  ? 

(a)  x-y  +  kz  =  0,x  +  y-z/k  =  {i\  (6)  3  a;  -  4  y  ^  A:,  (3  a;+4  y)k  =  \  ; 
(c)  X  —  y  +  3  A-^  =  3  A;,  k{x  -\-y)—  z  =  3. 

8.  Show  that  every  generating  line  of  the  hyperbolic  paraboloid 
or^/cfi  —  y'^b^  =  2  cz  is  parallel  to  one  of  the  planes  x^/a^  —  y^/b^  =  0. 

380.  Surfaces  in  General.  When  it  is  required  to  deter- 
mine the  shape  of  a  surface  from  its  cartesian  equation 

the  most  effective  methods,  apart  from  the  calculus,  are  the 
transformation  of  coordinates  and  the  taking  of  cross-sections, 
generally  (though  not  necessarily  always)  at  right  angles  to 
the  axes  of  coordinates.  Both  these  methods  have  been  ap- 
plied repeatedly  to  the  quadric  surfaces  in  the  preceding 
articles. 

381.  Cross-Sections.  The  method  of  cross-sections  is  ex- 
tensively used  in  the  applications.  The  railroad  engineer  de- 
termines thus  the  shape  of  a  railroad  dam ;  the  naval  architect 
uses  it  in  laying  out  his  ship ;  even  the  biologist  uses  it  in  con- 
structing enlarged  models  of  small  organs  of  plants  or  animals. 

382.  Parallel  Planes.  When  the  given  equation  contains 
only  one  of  the  variables  x,  y,  z,  it  represents  of  course  a  set  of 
parallel  planes  (real  or  imaginary),  at  right  angles  to  one  of 
the*  axes.     Thus  any  equation  of  the  form 

F{x)=0 

represents  planes  at  right  angles  to  Ox,  of  which  as  many  are 
real  as  the  equation  has  real  roots. 


XVI,  §  386]  QUADRIC  SURFACES  351 

383.  Cylinders.  When  the  given  equation  contains  only  two 
variables  it  represents  a  cylinder  at  right  angles  to  one  of  the 
coordinate  planes.     Thus  any  equation  of  the  form 

F{x,y)=0 
represents  a  cylinder  passing  through  the  curve  F{x,  y)  =  0  in 
the  plane  Oxy,  with  generators  parallel  to  Oz.    If,  in  particular, 
F(x,  y)  is  homogeneous  in  x  and  y,  i.e.  if  all  terms  are  of  the 
same  degree,  the  cylinder  breaks  up  into  planes. 

384.  Cones.  When  the  given  equation  F(x,  y,  2)=0  is 
homogeneous  in  x,  y,  and  z,  i.e.  if  all  terms  are  of  the  same 
degree,  the  equation  represents  a  general  cone,  with  vertex  at 
the  origin.  For  in  this  case,  if  {x,  y,  z)  is  a  point  of  the  sur- 
face, so  is  the  point  (lex,  ky,  kz),  where  k  is  any  constant;  in 
other  words,  if  P  is  a  point  of  the  surface,  then  every  point  of 
the  line  OP  belongs  to  the  surface ;  the  surface  can  therefore 
be  generated  by  the  motion  of  a  line  passing  through  the  origin. 

385.  Functions  of  Two  Variables.  Just  as  plane  curves  are 
used  to  represent  functions  of  a  single  variable,  so  surfaces  can 
be  used  to  represent  functions  of  two  variables.  Thus  to  obtain 
an  intuitive  picture  of  a  given  function  f{x,  y)  we  may  con- 
struct a  model  of  the  surface 

such  as  the  relief  map  of  a  mountainous  country.     The  ordi- 
nate z  of  the  surface  represents  the  function. 

386.  Contour  Lines.  To  obtain  some  idea  of  such  a  surface 
by  means  of  a  plane  drawing  the  method  of  contour  lines  or 
level  lines  can  be  used.  This  is  done,  e.g.,  in  topographical 
maps.  The  method  consists  in  taking  horizontal  cross-sections 
at  equal  intervals  and  projecting  these  cross-sections  on  the  hori- 
zontal plane.  Where  the  level  lines  crowd  together  the  surface 
is  steep ;  where  they  are  relatively  far  apart  the  surface  is  flat. 


352 


SOLID  ANALYTIC   GEOMETRY       [XVI,  §  386 


EXERCISES 

1.    What  surfaces  are  represented  by  the  following  equations  ? 


(a)  Ax-{-By+C  =  0. 

(c)  y^-\-z^  =  a^. 

(e)  zx  =  a^. 

(g)  x^-Sx^-x+S  =  0. 

(0  y  =  x^  -  X  -  e. 

(k)  x^  +  2  ?/2  =  0. 

(m)  x'^-y^  =  z^ 

(0)   (x-l){y-2)(z-S)  =  0. 


(b)  xcos^  -\-  ysinp  =p. 

(d)  z^-x^  =  a^ 

(/)  z^  =  4ay.^ 

(h)  xyz  =  0, 

U)  yz^-9y  =  0. 

(I)  a;2  =  yz. 

(n)  y2  +  2z'^-\-4zx  =  0. 

(p)  a;3  -f  y3  —  3  xyz  =  0. 


2.  Determine  the  nature  of  the  following  surfaces  by  sketching  the 
contour  lines : 

(a)  z=:x-{-y.         (b)  z  =  xy.  (c)z  =  y/x.  (d)  z  =x^ -^y^. 

(e)  z=x^-y^-\-4.     (f)z  =  x^.  (g)  z=x'^-\-y^-4:X.     (h)z  =  xy-x. 

{i)  z  =  2\  (j)  y=z'^-ix.     {k)y  =  Sz^  +  x^.       {l)z=nx+y'\ 

3.  The  Cassinian  ovals  (§  270)  are  contour  lines  of  what  surface  ? 

4.  What  can  be  said  about  the  nature  of  the  contour  lines  of  a  sur- 
face z  =f{x)  ?  Discuss  in  particular  :  (a)  z  =  x^  —  9  ;  (b)  z  =  x^  —  8  ; 
(c)  y  =  z^  +  2z. 

387.  Rotation  of  Coordinate  Trihedral.     To  transform  the 

equation  of  a  surface  from  one  coordinate  trihedral  Oxyz  to  another 
Ox'y'z',  with  the  same  origin  O,  we 
must  find  expressions  for  the  old  co- 
ordinates X,  y,  z  of  any  point  P  in  terms 
of  the  new  coordinates  x',  y',  z'.  We 
here  confine  ourselves  to  the  case  when 
each  trihedral  is  trirectangular ;  this  is 
the  case  of  orthogonal  transformation, 
or  orthogonal  substitution. 

Let  li,  wi,  wi,  be  the  direction  cosines 
of  the  new  axis  Ox'  with  respect  to  the 
old  axes  Ox,  Oy,  Oz  (Fig.  150) ;  similarly 
h,  m2,  W2  those  of  Oy',  and  Z3,  ma,  W3  those  of  Oz'. 
the  scheme 


Fig.  150 
This  is  indicated  by 


XVI,  §389]  QUADRIC  SURFACES  353 


x' 

y' 

0' 

h 

h 

h' 

Wli 

m2 

mz 

ni 

W2 

tiz 

which  shows  at  the  same  time  that  then  the  direction  cosines  of  the  old 
axis  Ox  with  respect  to  the  new  axes  Ox',  Oy' ^  Oz'  are  Zi,  ^2,  h,  etc. 

388.  The  nine  direction  cosines  h,  h,  •••  n^  are  sufficient  to  determine 

the  position  of  the  new  trihedral  Ox'y'z'  with  respect  to  the  old.     But 

these  nine  quantities  cannot  be  selected  arbitrarily  ;  they  are  connected  by 

six  independent  relations  which  can  be  written  in  either  of  the  equivalent 

forms 

h^  +  wii2  +  n{^  =  1,  ^2^3  +  m^mz  +  ruiiz  =  0, 

(1)  ^2^  +  ^2'-^  +  «2^  =  1,  hh -h  ms^ni  +  n^ni  —  0, 

h^  +  rriz^  +  W32  =  1,  hh  +  Wim2  +  W1W2  =  0, 
or 

h^  +  h^  +  ?3^  =  1,  wini  +  W2W2  +  wisws  =  0, 

(1')        wii2  +  W22  +  W32  =  1,  mh  +  n2h  +  nsh  =  0, 

Wl^  +  W2^  +  W32  =  1 ,  ZlWi  +  l2'm2  +  ?3W»3  =  0. 

The  meaning  of  these  equations  follows  from  §§  297  and  300.  Thus 
the  first  of  the  equations  (1)  expresses  the  fact  that  h,  mi,  ui  are  the 
direction  cosines  of  a  line,  viz.  Ox'  ;  the  last  of  the  equations  (1')  ex- 
presses the  perpendicularity  of  the  axes  Ox  and  Oy  ;  and  so  on. 

389.  If  X,  y,  z  are  the  old,  x',  y',  z'  the  new  coordinates  of  one  and 

the  same  point,  we  find  by  observing  that  the  projection  on  Ox  of  the 

radius  vector  of  P  is  equal  to  the  sum  of  the  projections  on  Ox  of  its 

components  x',  y',  z'  (§  294),  and  similarly  for  the  projections  on  Oy 

and  Oz  : 

X  =  hx'  +  hy'  +  hz', 

(2)  y  =  mix'  +  mzy'  +  W30', 
z  =  mx'  +  n^y'  +  n^z'. 

Indeed,  these  relations  can  be  directly  read  off  from  the  scheme  of 
direction  cosines  in  §  387. 

Likewise,  projecting  on  Ox',  Oy',  Oz',  we  find 

x'  =  hx  +  miy  +  niz, 
(2')  yi  =  I2X  +  TO22/  -I-  n2Z, 

z'  =  Izx  +  m^y  +  mz. 
2a 


354  SOLID  ANALYTIC  GEOMETRY      [XVI,  §  389 

As  the  equations  (2),  by  means  of  which  we  can  transform  the  equation 
of  any  surface  from  one  rectangular  system  of  coordinates  to  any  other 
with  the  same  origin,  give  x,  y,  z  as  linear  functions  of  x',y',  z',  it  follows 
that  such  a  transformation  cannot  change  the  degree  of  the  equation  of 
the  -surface. 

390.  Th-fe  equation  (2')  must  of  course  result  also  by  solving  the  equa- 
tions (2)  for  x',  y',  z',  and  vice  versa.    Putting 

h      h      h 
nil     Wi2    wi3    =  D, 
ni      Ui      nz 
solving  (2)  for  x',  y\  z',  and  comparing  the  coefficients  of  x,  y,  z  with 
those  in  (2')  we  find  the  following  relations  : 

I)h  =  m^nz  —  W3W2,      Bmi  =  n^h  —  Ush,      Dni  =  hm^  —  hm^i     etc. 

Squaring  and  adding  the  first  three  equations  (compare  Ex.  3,  p.  45) 
and  applying  the  relations  (1)  we  find  :  D^  =  \. 

By  §  321,  D  can  be  interpreted  as  six  times  the  volume  of  the  tetrahe- 
dron whose  vertices  are  the  origin  and  the  points  x',  y',  z'  in  Fig.  150,  i.e. 
the  intersections  of  the  new  axes  with  the  unit  sphere  about  the  origin. 
The  determinant  gives  this  volume  with  the  sign  +  or  —  according  as  the 
trihedral  Ox'y'z'  is  superposable  or  not  (in  direction  and  sense)  to  the 
trihedral  Oxyz  (see  §  391).     It  follows  that  D  =±1  and 

h  =  ±  {rmnz  —  mz7i2),     mi  =  ±  {n2h  —  n^h),     Wi  =  ±  (^2^3  —  ^3^2), 

?2  =±  (W3W1  —  miWs),      W2  =±  (WsZi  -  W1Z3),      W2  =  ±  (^3^11  —  Zim3), 

l3=±  (miW2  — m2ni),    W3  =±  (nih  —  W2?i),     m  =±  {hm2—  hmi), 
the  upper  or  lower  signs  to  be  used  according  as  the  trihedrals  are  super- 
posable or  not. 

391.  A  rectangular  trihedral  Oxijz  is  called  right-handed  if  the  rotation 
that  turns  Oy  through  90°  into  Oz  appears  counterclockwise  as  seen  from 
Ox  ;  otherwise  it  is  called  left-handed.  In  the  present  work  right-handed 
sets  of  axes  have  been  used  throughout. 

Two  right-handed  as  well  as  two  left-handed  rectangular  trihedrals  are 
superposable ;  a  right-handed  and  a  left-handed  trihedral  are  not  super- 
posable. The  difference  is  of  the  same  kind  as  that  between  the  gloves 
of  the  right  and  left  hand. 

Two  non-superposable  rectangular  trihedrals  become  superposable  upon 
reversing  one  (or  all  three)  of  the  axes  of  either  one. 


XVI,  §  393]  QUADRIC  SURFACES  355 

392.  The  fact  that  the  nine  direction  cosines  are  connected  by  six  rela- 
tions (§  388)  suggests  that  it  must  be  possible  to  determine  the  position  of 
the  new  trihedral  with  respect  to  the  old  by  only  three  angles.  As  such 
we  may  take,  in  the  case  of  superposable  trihedrals,  the  angles  0,  0,  \}/, 
marked  in  Fig.  160,  which  are  known  as  Eulefs  angles. 

The  figure  shows  the  intersections  of  the  two  trihedrals  with  a  sphere 
of  radius  1  described  about  the  origin  as  center.  If  OiVis  the  intersection 
of  the  planes  Oxy  and  Ox'y',  Euler's  angles  are  defined  as 

d  =  zOz',    <t>  =  NOx',    \p  =  xON. 

The  line  ON  is  called  the  line  of  nodes^  or  the  nodal  line. 

Imagine  the  new  trihedral  Ox'y'z'  initially  coincident  with  the  old 
trihedral  Oxyz,  in  direction  and  sense.  Now  turn  the  new  trihedral 
about  Oz  in  the  positive  (counterclockwise)  sense  until  Ox'  coincides  with 
the  assumed  positive  sense  of  the  i^odal  line  ON;  the  amount  of  this 
rotation  gives  the  angle  \^.  Next  turn  the  new  trihedral  about  ON  in  the 
positive  sense  until  the  plane  Ox'y'  assumes  its  final  position  ;  this  gives 
the  angle  6  as  the  angle  between  the  planes  Oxy  and  Ox'y\  or  the  angle 
zOz'  between  their  normals.  Finally  a  rotation  of  the  new  trihedral 
about  the  axis  Oz\  which  has  reached  its  final  position,  in  the  positive 
sense  until  Ox'  assumes  its  final  position,  determines  the  angle  0. 

393.  The  relations  between  the  nine  direction  cosines  and  the  three 
angles  of  Euler  are  readily  found  from  Fig.  150  by  applying  the  fundamen- 
tal formula  of  spherical  trigonometry  cos  c  =  cos  a  cos  6  +  sin  a  sin  h  cos  7 
successively  to  the  spherical  triangles 

xNx'^    xNy',    xNz'^ 
yNx',    yNy',    yNz', 

zNx'^     zNy'^     zNz'. 
We  find  in  this  way  : 

li  =  cos  xj/  cos  0  —  sin  ^  sin  (p  cos  6, 
mi  =  sin  \{/  cos  <f)  -f  cos  xj/  sin  0  cos  ^, 
wi  =  sin  0  sin  0, 

?2  =  —  cos  i/'  sin  0  —  sin  \//  cos  0  cos  0,  Z3  =  sin  ^  sin  0, 

m2=—  sin  1^  sin  0  +  cos  ^  cos  0  cos  0,  mz=—  cos  \p  sin  0^ 

ni  =  cos  0  sin  0,  m  =  cos  0. 


APPENDIX 

NOTE    ON  ABRIDGED    NUMERICAL    MULTIPLICATION 
AND   DIVISION 

1.  In  multiplying  two  numbers  it  is  convenient  to  write  the 
multiplier  not  below  but  to  the  right  of  the  multiplicand  in 
the  same  line  with  it,  and  to  begin  the  formation  of  the  par- 
tial products  with  the  highest  figure  (and  not  with  the  lowest). 
The  most  important  part  of  the  product  is  thus  obtained  first. 
The  partial  products  must  then  be  moved  out  toward  the  right 
(and  not  to  the  left).     Thus : 


35702 


285616 
24991 4 
71 
17 


87025 


404 
8510 


310696  6550 

2.  "Long"  multiplications  like  the  above  rarely  occur  in 
practice.  Generally  we  have  to  multiply  two  numbers  known 
only  approximately,  to  a  certain  number  of  significant  figures. 
Suppose  we  want  to  find  the  product  of  3.5702  and  8.7025,  five 
significant  figures  only  being  known.  It  is  then  useless  to 
calculate  the  figures  to  the  right  of  the  vertical  line  in  the 
scheme  above.  To  omit  this  useless  part  we  proceed  as  fol- 
lows. In  multiplying  by  8,  place  a  dot  over  the  last  figure  2 
of  the  multiplicand ;  in  multiplying  by  7,  place  a  dot  over  the 
0  of  the  multiplicand,  beginning  the  multiplication  with  this 
figure  (adding,  however,  the  1  which  is  to  be  carried  from  the 
preceding  product  7x2);  then  to  indicate  the  multiplication 
by  0  simply  place  a  dot  over  the  7  of  the  multiplicand;  the 

356 


APPENDIX 


357 


multiplication  by  2  has  then  to  begin  at  the  5  of  the  multipli- 
cand.    Thus  we  obtain : 

3.5702 1  8.7025 
28  5616 
2  4991 
71 

18 

31.0696 
The  last  figure  so  found  is  slightly  uncertain,  just  as  the 
last  figures  of  the^given  numbers  generally  are. 

3.   In  division  it  is  most  convenient  to  place  th^^ivisor  to 
the  right  of  the  dividend.     Thus 

3.1416  =-8.90702 


27.9823 
25  1328 


2  8495 
2  8274 


220 
219 


0 
4 

600 
912 


68800 
62832 


To  cut  off  the  superfluous  part  to  the  right  of  the  vertical 
line,  subtract  the  first  partial  product  as  usual ;  then  cut  off 
the  last  figure  from  the  divisor  and  divide  by  the  remaining 
portion ;  go  on  in  this  way,  cutting  off  a  figure  from  the  divisor 
at  every  new  division  until  the  divisor  is  used  up.     Thus : 
27.9823  |3J^  =  8.90701 
25  1328 
2  8495 
2  8274 
221 
220 
1 


ANSWERS 

[Answers  which  might  in  any  way  lessen  the  vahie  of  the  Exercise  are  not 

given.] 

Pages  9-10.     5.   2|  miles.  16.  173.9  ft. 

Pages    13-14.     3.  22.  4.  ^(pc  +  c«  +  ab). 

7.  K«^  +  2  6c  -  2  ca  -  b'^)  =  K«  -  &)(«  +  b  -2c). 

Pages  17-18.      4.  |rir2  sin  (02  -  0i)- 

5.  I[r2r3  sin  (03  -  02)  +  nn  sin  (0i  -  03)  +  rir2sin  (02-  0i)]. 

6.  — ^'^^^cosi^(02  —  0i).  7.     rcos0  =  X  +  y  cos  w,  rsin0  =  ysin  w. 
•J'l  +  r2 

Page  22.    17.  They  intersect  at  [\(xi-\-X2-{-x^  +  Xi),  ^(^1+^2+2/3+2/4)]. 
20.  [i(a;i  +X2  +  X3),  i(yi  +  2/2  +  2/3)]. 

Page  35.     21.  P  =  1000(1  +  r)  ;  P  =  1000  +  60  n. 

Page  38.     14.  No. 

Page  45.     1.  (e)  sin2j3;   (/)  a2a3  +  «3«i  +  «ia2. 

8.  (&)   (4,  3),  (4,  -  3),   (-  4,  3),  (-  4,  -  3)  ;    (d)  (3,  -  2)  ; 
(e)  (±i,  ±3);    (/)   (t,  i)-. 

Pages  48-49.     1.   (a)  0  ;  (6)  0  ;  (c)    -113;  (d)   -5;  (e)  1. 

4.    (a)   (2,  -  1,  3)  ;   (6)  (83/41,  -  81/41,  -  35/41)  ;  (c)  (-  5,  3,  -  2)  ; 
id)   (±3,  i2,  ±4);  (e)  (±1,  ±1,  ±1);  (/)  (1,0,-3). 

Page  53.     1.   (a)    0  ;  (6)   -  180;    (c)   -27846;  ((?)  7728;  (e)  36; 
(/)  550. 

Page  57.     6.    (27/2,  -77/2). 

Pages  59-60.     6.640/39.         9.    (&1W2  -  62^1)2/2  mim2(wii  -  m2). 
10.  (3,  i). 

Pages  65-66.     2.  (a)  r  sin  0  =  ±  5  ;   (6)  r  cos  0  =  ±  4  ; 
(c)  rcos(0- f  7r)=  ±  12. 

3.  0  =  0,  rsin  0  =  9,  0  =  ^  TT,  r  cos  0  =  6.      14.  8464/85. 
19.    (-  5,  -  10).        21.  X  =  1  (by  inspection),  4x  -  3y -\- 16  =0. 

359 


360  ANSWERS 


Page  68.     4.  K^  -  ab  =  0. 

< 


Page  69.     1.   tan-i  ^^^'  ~  ^^  ;  a  =  -b,  h^=ab. ' 
a  -i-  b 

4.  [mi(62  —  &)— W2(&i  —  6)]^/2mim2(m2  — «ii). 

6.  r(2  cos  0  -  3  sin  0)  +  12  =  0. 

10.  1  hr.  10  m.  ;  176  miles  from  Detroit. 

Page  75.     6.  560.        7.  120.        8.  65200.        9.  60  ;  24,  36. 

10.  487635,  32509,  1653. 

11.  „C'i„,  when  n  is  even  ;  „C'|(^_j^  =  «^^cn+i)'  ^^^"  "  ^^  ^^^* 

12.  66."       13.  120. 

Pages  82-83.     2.    aox^  +  aix^  +  a2X  +  as-        4.    8  abed. 

6.  (a)  a;  =  2,  ?/=-!,  0=2,  10  =  3;   (&)  x  =  1,  y  =  3,  5r  =  2,  lo  =  -  1. 

7.  (a)  No;  (b)  Yes. 

8.  cos^  a  +  cos2  /3  +  cos'^  y  +  2  cos  «  cos  /3  cos  7  pi- 
pages 85-86.    2.   (a)  ABG+2FGH-AF^-BG-^-CH^] 

(b)  x^  +  i/2  +  0-^  -  2(2/5  +  0X  +  xy)  ;    (c)  -  (x^  +  i/^  +  0^)  ;    (e)  4. 

7.  («)  l+a2  +  62  +  c2.  ^5)  (a(^  +  c/- &e)2  ;  (c)   (a(?  +  fte  +  c/)'^. 

Pages  90-91.     6.   x'^  +  tf  -  96x- 6iy  ■{■  2408  =  0  ;  31.8  ft.  or  66.3  ft. 

8.  a;2  +  ?/2  -  16  X  +  8  ?/  +  60  =  0.  9.    A    circle    except    for    k  =±l. 

10.    a;2  _|.  y2  _,_  4  L±A%  4- 4  =  0. 
1  —  k^ 

Page  92.     2.    (a)  7-2-20  r  sin  0+75=0  ; 
(6)  f^  -12  r  cos  (0  -  i  tt)  4-  18  =  0  ;    (c)    r  +  8  sin  0  =  0. 

Page  94.     8.   ^2  -  6  a:  +  28  =  0.  9.   x'^  +  2  pwx  +  gm^  =  0. 

Page  96.     3.   (-6,  -1),  (29/106,  42/53). 
7.    8a;-4?/-ll±15\/2  =  0.    t^ 

Page  98.     3.    (xi  -  h)  {x  -  h)  +  {yi  -  k)  (y  -  k)  =  r^. 

7.  i-r^A/C-rW/C),        8.    (2,1). 

Page  100.     6.  {x  -  79/38)2  +  (y  -  55/38)2  =  (65/38)2. 

8.  ^2  +  ?/2  +  4  X  —  2  ?/  -  15  =  0. 

Page  105.     1.    (c)  Polar  lies  at  infinity. 

Pages  108-109.     3.  Let  L,  M  be  the  intersections  of  the  circle  with 
CPi,  then  ^2  ~  r2  =  LPi  -  MPi. 


ANSWERS  361 


6.    (c)  2x24-2y2_|.22x+6?/+15=0,  2x2+2?/2_i0x-10?/-25=0. 

12.  If  the  vertices  of  the  square  are  (0,  0),  (a,  0),  (0,  a),  (a,  a)  and  A;2 
is  the  constant,  the  locus  is  2  x2  +  2  ?/2  —  2  ax  -  2  ai/  +  2  a2  -  ^2  =  0  ; 
^•>a;  |aV6. 

13.  If  the  vertices  of  the  triangle  are  (a,  0),  (—a,  0),  (0,  aV3)  and 
A;2  is  the  constant,  the  locus  is  3  x2  +  3  y2  _  2  VS  a?/  +  3  a2  -  2  A:2  =  0. 

Page  126.      8.    (a)   (3  +  4i)/25;     (&)   (3  +  VrO/14  ; 
(c)   (-6  +  30/34;     (c?)    (1-6  0/37. 

Page  130.    7.   (^)  ±K^^+v'20;  {h.)   v^2(cos80°+isin80°), 
\/2(cos  200°  +  i  sin  200=),  ^2(cos  320°  +  i  sin  320°). 

Pages  135-136.     10.    (a)  2  ?/  =  3  x2  +  5  x  ; 

(6)  12  ?/  =  -  5  x2  +  29  X  -  18. 

11.  300  y  =  -  x2  +  230  x  ;    44.1  ft.  above  the  ground ;    230  ft.  from  the 

starting  point. 

20.   (6)  No  parabola  of  the  form  y  =  ax^  +  6x  +  c  is  possible. 

Page  138.     13.    (2,3),  (-1.8,3.6),  (3.1,  -2,8),  (-3.3,-3.8). 

Page  142.     6.   East,  East  33°  41'  North,  East  53°  8'  North,  East  18° 
26'  South. 
10.    100/(9r+4). 

Pages  145-146.     10.   0,  8°  8'.         11.    7°  29'. 
15.    When  the  side  of  the  square  is  3  in. 
18.    (a)  6  y  =  x8  +  6  2ic  -  19  X  ;    (6)  7  y  =  2  x^  -  x2  -  29  x  +  36. 

Page  147.     1.   (a)   -1,  3.62,  1.38;  (6)  -1.45,  -.403,  .855  ; 
(c)  -1.94,  .558,  1.38;  (d)  2.79. 

Page  154.     4.    (d)   -252xM;  (d)  ^0  a%^  -  SO  a^b^  ;  (h)  27/a25. 

Page  159.     3.    (a)  PiP2=Ps;  (&)  Pi^Ps=P2^  ;  (c)  pi^=27 p2^=7292-)s\ 

Page  162.     1.    -  1.88,  1.53,  .347. 

Page  167.     1.     (a)  4.06155 ;     (b)   ±2.08779;     (c)   1.475773. 
2.   2.0945514.        3.    .34899. 

4.  (a)   (1.88,  3),  (-  1.53,  3),  (-  .347,  3)  ; 

(b)  (.309,  1.10),  (1.65,  1.55),  (-1.96,  .347)  ;  (c)  (-2.106,  -1.0266). 

5.  3.39487  in.       6.   9.69579  ft.        7.    -  2,  1  ±  VS. 
8.    .22775,  3.1006.        9.    5.4418  ft. 

10.  (2,  3),  (-  1.848,  3.584),  (3.131,  -  2.805),  (-  3.383,  -  3.779). 

11.  (2.21,  .89).         12.    .34729  a. 


362  ANSWERS 

Pages  173-174.      2.    (a)   (4,  i  tt),  (4,  |  tt)  ;    (b)  (a,  ^  tt),  (a,  |  tt)  ; 
(c)(4,  0);  (d)  (4aa'r),  (4  a,  fTr). 

7.  (a)  2/2  -  4  x  +  4  =  0  ;  (6)  14  2/2  -  45  X  +  52  ?/  +  60  =  0. 

8.  (6)  a;2  -  10  a;  -  3  2/  +  21  =  0  ;  (c)  ^2  +  2  a;  +  y  -  1  =  0. 

9.  The  equation  of  a  parabola  contains  an  xy  term  when  its  axis  is  oblique 
to  a  coordinate  axis. 

Pages  179-180.     1.    (a)  18  a;  -  30  ; 

(5)  6  x5  -  30  X*  +  48  x3  -  24  ic2  +  8  X  -  8. 

2.  (a)   y'=5/2y;    (b)  y' =  6/(5  -  2  y)  ;    (c)  y'  =  2/Sy. 
5.    (a)y'=-y/x;    (b)  y' =  (6  -  2  xy) /x^  ; 

(c)  2/'=-(^x  +  iry  +  G^)/(£rx  +  5y  +  i?').- 

Pages  186-188.     8.   («)  2/=0  ;    (6)  2a:+2i/-9=0,  2a;-|/-18=0; 

(c)  2  X  +  2  y  -  9  =  0,  8  X  +  16  y  -  27  =  0,  24  X  -  16  2/  -  153  =  0  ; 

(d)  8  X  -  16  2/  -  27  =  0. 

14.    Directrix.         15.   y'^  =  a{x-3a).         22.   1^(1  + m2). 

m2 

29.  a:2-80x-2400?/  =  0;    0,  -  |,  -  |,  -  i  0,  f,  2. 

30.  x2  =  360(2/ -20). 

Pages  194-195.     2.    (3  7r-4)/6  7r.       3.    §  a2  (1±!^. 

8.    («)  64/3;    (6)  625/12;    (c)   1/12.       9.    123.84  ft^.       10.    1794i  tons. 
11.    199.4  ft2. 

Page  197.     To  obtain  the  following  solutions,  take  the  origin  at  one 
end  of  the  beam  and  the  axis  Ox  along  the  beam. 

1.    F-  W,  M=  W(x-l).  2.    F=io(^l-x),  M:=  ^w(l-x)x, 

3.  (a)  Fi=-wx,  Mi  =  -^wx^;  F2=w{ll—x),  M2=-iw{^P—lx+x^); 
Fi  =  w{l-x),  Mz=-\w{l-xy) 

{b)  Fi=-W,  Mi=-Wx',  F2  =  0',  M.2=-\Wl] 
Fs=  W,M3=-W(il-x). 

4.  (a)  Fi  =  lwl,  Mi  =  lwlx',   F2  =  io(^l  —  x), 
M2=-^  w?(a;2  -lx-\-\l^);  i^s  =  -  i  wjZ,  ^3  =  z '^K^  -  ^)' 

Page  200.     9.   8^2  -  2  xy  +  8  y2_  63  =  0. 

Page  204.     10.  3  x2  -  ?/2  =  3  a^.        11.   b.        14.    2  xy  =  1. 

Pages  211-212.     2.  ^X4-^V=c2.     13.  54.5  ft.,  42.2  ft.     18.62/^2. 
x  y 

23.   An  ellipse  or  hyperbola  according  as  one  circle  lies  within  or  without 
the  other  circle. 


ANSWERS  363 

Pages  221-222.     7.    (a)  A'a^  -  B^b^  =  C^ ; 
(&)   rt2cos2/3-  &2sin2/3=i)2. 
19.    62,        21.   a2  +  62  .  ^2  _  62. 

22.   4ab.  23.   sin-i  (a6/a'6')- 

25.    (a)  a;2  +  2/2  =  oj2  +  62  ;    (&)  ^2  +  2/2  =  ^2  _  ^2. 

Page  22?!     3.    (a)  (1,  -1),  (1±V2,  -1),  a;=li|V2; 
(&)  (i,0),  (1,0),  (-|,0),:r.  =  0,a;  =  l. 
4.   2  62/a.        8.    (a)  a2j,2  ^^  ft2j;(«  _  a^) ;  (&)  b'^x^  =  a^y{b  -  y). 

10.  Two  straight  lines. 

Page  235.     2.    (a)   Vertices    (5,    3),    (8,    3);     semi-axes  3/2,    V2. 
(6)  Vertices  (4,   8/3),    (8,    8)  ;    semi-axes   10/3,    5\/3/3. 

(c)  vertices  (17/6,  7/5),  (1,  3)  ;  semi-axes  \/65/5,  \/l3/2. 

3.  3a:  +  2?/-2=0;  (21/13,  -37/26),  10/Vl3. 

Page  237.     5.    (acosd,  —  asind),  x^ -{- y'^  —  2  a(xcose  —  ysine)  =  0. 

Pages  246-247.     2.    (a)   3x-142/=0;    (b)  y  =  -S/l3,    x=-14/13. 
6.   2  x2  -  xy  -  15  2/2  -I-  X  +  19  2/  -  6  =  0, 
2  x2  -  iC2/  -  15  2/2  +  X  -^  19  2/  -  28  =  0. 
6.   6  x2  +  icj/  -  2  2/2  -  9  x  -t-  8  2/  -  46  =  0, 
6  a;2  -H  a;2/  -  2  y2  _  9  a;  +  8  2/  +  34  =  0. 

11.  (a)  a:2/4  +  y^  =  1  ;    (6)  x2/4  -  2/V2  =  1  ;  (c)  3  x2  +  ^2  _,_  ^  =  0  ; 
((?)  a;2/16  +_2/V4  =  1 ;      _(e)   (3  +  Vl7)x2  -}-  (3  -  Vl7)2/2  =  4  ; 

(/)  (2-hV2)x2-H(2-V2)y2  =  l. 

15.   x^  +  yi=  a^. 

19.   Equilateral  hyperbola. 

Page  253.     2.    («)  Simple  point ;     (6)  node ;     (c)  cusp ;     (d)  cusp. 

4.  (a)  None ;         (6)  node  at  (6,  0)  ;         (c)  isolated  point  at  (a,  0)  ; 

(d)  cusp  at  (a,  0). 

Page  260.    4.  r  =  a(sec  <f>  ±  tan  0)  or  (x  —  a)y2  _^  3.2(-a.  ^_  q,)  _  q. 
10.   x22/2  zz  a2(a;2  4.  ^2).        11.    Cissoid  (a  -  x)y^  =  x^. 

12.  2/(x2-l-2/2)  =  a(x2-2/2).        13.   r  =  actn0. 
14.    (x2  -H  2/2)2  _  4  «a;(x2  -  2/2). 

Page  283.       6.  ^  +  ^^  etc. 

V2(l  +  ZZ' -l-mm' +  7i«') 
13.    ^  (xi  +  X2  +  X3),  i  (yi  +  ^2  +  ys),  i  (^1  +  2^2  +  Z3). 

Page  287.     6.    cos-i  (7/3V29). 


364 


ANSWERS 


Page  291.     2.    ^V465. 


3. 


^269. 


6.  (3962,  47^  43',  276°  16'),  (320,  -  2914,  2666),  2931. 

7.  I  riViVl  —[cos  di  cos  62  +  sin  Oi  sin  62  cos  (<pi  —  02)]"^ 


02)+  cosfli  cos  ^2]. 


8.    y/ri^  +  rz^  —  2  rir2  [sin  di  sin  ^2  cos  (0i 
10.    -  1,  10,  7. 

Page  296.     3.   39  a:  -  10  ?/  +  7  0  -  89  =  0. 

5.  97/28,  -  97/49,  -  97/9.        7.    Sx  -  4:y  +  2  z- 6  =  0. 

Page  300.     5.  4ic  +  8?/  +  2;  =  81,  4x-|-8a;  +  ^  =  90. 

Page  303.     2.    (a)  56/3;  (ft)  0;  (c)  19/3. 

Page  306.      12.   Sx-2y  =  l.        13.   6a;  +  11  ?/ +  90  =  58. 
16.    70°  31'.        17.    cos-i(2/i-^  +  3rt2)/(4/i2_|.3«2). 

Pages  314-316.     3.   69°  29'.         19.    (a)   V637l9;   (ft)  V194/33. 
21.    X  -  2  y  +  ^  +  8  =  0. 

X2  —  Xi    2/2  —  2/1     ;22  -  i^l 
24.  ai  fti  ci      =0. 

a2  ft2  C2 

Page  320.     11.    ( -  3,  -  3,  2),  (9,  9,  -  6). 

Pages325-326.     4.    (1,  0,  -  3),  (-  9/11,  20/11,  27/11). 

7.  a:2  -  3  y^  -  S  z^  =  0.         13.   25(^2  +  y^  +  z^)  =  16^,  25  0  =  64. 

Pages  329-331.     4.    (4,-5,-3).         5.    (4,6,2). 

6.  ^x  +  2y  -  z  =  25,2x~3y  -\-z  +  25  =  0. 

20.  9ic2  +  4?/2  +  1.3 2;2  +  2 X2/  -  273  =  0. 

21.  (x  -  ZA:)2  +  (^  -  mky^  +  (s  -  wA;)2  =  r2. 

22.  ll{x-\-h)+m{y-\-j)  +  n{zi-k)Y-[{x+hy-\-{y-hj)^+(z  +  k)^-r'2]=0. 

Page  336.     3.    Va^  -  c2  x  ±  Vft2  -  c2  0  =  0. 

6.     (a;2  +  ^2  +  ;22  _  «2  _  ft2)2  _  4  52(q;2  _  ^2)  ^  Q. 

8.  (a)  16a2(a;2  4-;32)  =  y4.  (&)  16  a2[(x  +  a)2  +  02]  =  (4a2  -  y2)2. 

9.  y^{x^  +  z^)=a^.  ^ 


INDEX 


{The  numbers  refer  to  the  pages.) 


Abscissa,  1,  4. 

Absolute  value,  124. 

Acnode,  252. 

Adiabatic  expansion,  276. 

Algebraic  curves,  249-253. 

Amplitude,  16,  124. 

Angle  between  line  and  plane,  312 ; 

between  two  lines,   58,  284,   311; 

between  two  planes,  299. 
Anomaly,  16. 
Area   of   ellipse,    221 ;     of   parabolic 

segment,  191-195;   of  triangle,  11, 

12,  56,  288 ;  under  any  curve,  193. 
Argument,  124. 
Associative  law,  110. 
Asymptotes,  203. 
Axes  of  coordinates,  4,  277  ;  of  ellipse, 

198 ;   of  hyperbola,  202. 
Axis,  18 ;    of  parabola,  132,  170 ;    of 

pencil,  303  ;   of  symmetry,  137. 
Azimuth,  16. 

Bending  moment,  196-197. 
Binomial  coefficients,   152-154 ;    the- 
orem, 152-154. 
Bisecting  planes,  299. 
Bisectors  of  angles  of  two  lines,  64. 

Cardioid,  255. 

Cartesian  coordinates,  16. 

Cartesian  equation  of  conic,  225  ;  of 
ellipse,  199 ;  of  hyperbola,  202  ;  of 
parabola,  171. 

Cartesius,  17. 

Cassinian  ovals,  256,  259. 

Catenary,  188. 

Center  of  ellipse,  198,  215;  of  hyper- 
bola, 202,  215 ;  of  inversion,  101 ; 
of  pencil,  67;  of  sheaf,  304;  of 
symmetry,  137. 


Centroid,  22. 

Chord  of  contact,  103. 

Circle,  87-109  ;   in  space,  321. 

Circular  cone,  341. 

Cissoid,  255. 

Classification  of  conies,  225 ;  of 
quadric  surfaces,  342-345. 

Clockwise,  11. 

Cofactors,  52,  80. 

Colatitude,  290. 

Column,  41,  47. 

Combinations,  73-75. 

Common  chord,  107  ;  logarithms,  264. 

Commutative  law,  110. 

Completing  the  square,  88,  133. 

Complex  numbers,  100, 115,  117-130. 

Component,  19,  280. 

Conchoid,  254. 

Cone,  341,  351 ;  of  revolution,  342. 

Conic  sections,  223-231,  232. 

Conies  as  sections  of  a  cone,  228-231. 

Conjugate  axes,  327  ;  axis,  203  ;  com- 
plex numbers,  122  ;  diameters,  215- 
219 ;  elements  of  determinant, 
83;  lines,  327. 

Continuity,  155-156. 

Contour  lines,  351. 

Coordinate  axes,  4,  277 ;  planes,  277 ; 
trihedral,  277. 

Coordinates,  1,  5,  277 ;  polar,  16,  290. 

Cosine  curve,  261. 

Counterclockwise,  11. 

Cross-sections,  333,  337,  339,  350. 

Crunode,  252. 

Cubic  curves,  248 ;  equation,  146- 
147;   function,  143-147. 

Curve  in  space,  293. 

Cusp,  252. 

Cycloid,  257. 

Cylinders,  351. 


365 


366 


INDEX 


De  Moivre's  tlieorem,  126. 

Derivative,  139-141,  143,  149-152, 
177-179;  of  ax^,  139;  of  cubic 
function,  143  ;  of  function  of  a  func- 
tion, 178  ;  of  implicit  function,  177- 
179  ;  of  polynomial,  149-151 ;  of 
product,  178 ;  of  quadratic  func- 
tion, 140;   of  a:",  151. 

Descartes,  17. 

Determinant,  11,  13,  39;  of  n 
equations,  81 ;  of  order  n,  77 ;  of 
second  order,  41 ;  of  three  equa- 
tions, 48 ;  of  third  order,  47 ;  of 
two  equations,  41. 

Diameter,  333;  of  ellipse,  215;  of 
hyperbola,  218;  of  parabola,  184- 
185. 

Direction  cosines,  282,  307. 

Director  circle,  222 ;  sphere,  349. 

Directrices  of  conies,  223,  226. 

Directrix  of  parabola,  169. 

Discriminant  of  equation  of  second 
degree,  240-241 ;  of  quadratic 
equation,  92. 

Distance  between  two  points,  7,  17, 
278 ;  of  point  from  line,  63,  313 ; 
of  point  from  origin,  6,  278 ;  of 
point  from  plane,  298 ;  of  two 
lines,  313-314. 

Distributive  law,  110. 

Division,  abridged,  357. 

Division  ratio,  3,  8,  281. 

Double  point,  251. 

Eccentric  angle,  220. 

Eccentricity,  208,  223. 

Elements  of  determinant,  47;  of 
permutations  and  combinations,  70. 

Elimination,  43,  54,  82. 

EUipse,  198-222,  223,  229,  242-244. 

Ellipsoid,  332-334  ;  of  revolution,  334. 

Elliptic  cone,  341 ;  paraboloid,  340. 

Empirical  equations,  266-276. 

Epicycloid,  258. 

Equation  of  first  degree,  see  Linear 
equation  ;  of  line,  26,  32  ;  of  plane, 
293-297  ;   of  second  degree,  88. 

Equations  of  line,  308. 

Equator,  334. 

Equatorial  plane,  290. 

Equilateral  hyperbola,  203. 


Euler's  angles,  355. 
Expansion  by  minors,  51,  80. 
Explicit  and  implicit  functions,  177. 
Exponential  curve,  263. 

Factor  of  proportionality,  25. 
Factorial,  71. 

Falling  body,  15,  31,  69,  134. 
Family  of  circles,   107 ;    of  spheres, 

329. 
Foci  of  conic,  226;    of  ellipse,   198, 

223  ;  of  hyperbola,  201,  223. 
Focus  of  parabola,  169. 
Four-cusped  hypocycloid,  259. 
Function,  29  ;  of  two  variables,  351. 
Fundamental  laws  of  algebra,  110. 

Gas-meter,  27,  269. 

Gas  pressure,  272,  276. 

General  equation  of  second  degree, 

88,  233-247,  317,  342. 
Geometric  representation  of  complex 

numbers,  117. 

Higher  plane  curves,  248-276. 

Homogeneous  function  of  second 
degree,  241 ;  linear  equations,  43, 
54. 

Hooke's  law,  15,  25,  30,  38,  267, 
269. 

Horner's  process,  166. 

Hyperbola,  201-222,  223,  230,  242- 
244. 

Hyperbolic  logarithms,  264 ;  para- 
boloid, 340 ;   spiral,  259.  ^ 

Hyperboloid,  of  one  sheet,'  337-338 ; 
of  revolution  of  one  sheet,  338 ;  of 
revolution  of  two  sheets,  339 ;  of 
two  sheets,  338-339. 

Hypocycloid,  259. 

Imaginary  axis,  117;    ellipsoid,  339; 

numbers,    115;     roots,    127,    160; 

unit,  115 ;  values  in  geometry,  116. 
Implicit  functions,  177. 
Inclined  plane,  271. 
Induction,  mathematical,  71. 
Inflection,  144. 

Intercept,  26,  34  ;   form,  33,  295. 
Interpolation,  161. 
Intersecting  lines,  307. 


INDEX 


367 


Intersection  of  line  and  circle,   95 ; 

of  line  and  ellipse,  213  ;  of  line  and 

parabola,  181 ;    of  line  and  sphere, 

323 ;   of  two  lines,  39,  43. 
Inverse  of  a  circle,  101 ;    operations, 

111;     trigonometric    curves,    261- 

262. 
Inverses  of  involution,  112. 
Inversion,  100,  324. 
Inversions  in  permutations,  75. 
Inversor,  109. 
Irrational  numbers,  113. 
Isolated  point,  252. 

Latitude,  290. 

Latus  rectum  of  parabola,  170 ;  of 
conic,  224. 

Laws  of  algebra,  110;  of  exponents, 
112. 

Leading  elements,  83. 

Left-handed  trihedral,  354. 

Lemniscate,  257,  260. 

Level  lines,  351. 

LimaQon,  254. 

Limiting  cases  of  conies,  230. 

Line,  24,  307  ;  and  plane  perpendic- 
ular at  given  point,  312  ;  of  nodes, 
355  ;  parallel  to  an  axis,  23  ;  through 
one  point,  36,  308  ;  through  origin, 
24 ;  through  two  points,  36,  56, 
308. 

Linear  equation,  32,  293. 

Linear  equations,  n,  81 ;  three,  46, 
48,  302 ;   two,  39-42,  293,  302. 

Linear  function,  29,  131. 

Lituus,  259. 

Logarithm,  263-265. 

Logarithmic  paper,  274;  plotting, 
272-276. 

Longitude,  290. 

Major  axis,  199. 

Mathematical  induction,  71. 

Maximum,  141,  143. 

Measurement,  114. 

Mechanical  construction  of  ellipse, 
198;  of  hyperbola,  201;  of  parab- 
ola, 171. 

Melting  point  of  alloy,  175,  269. 

Meridian  plane,  290;  section,  335. 

Midpoint  of  segment,  9. 


Minimum,  141,  143. 

Minor  axis,  199. 

Minors  of  determinant,  51,  80. 

Modulus   of   complex  number,    124 ; 

of  logarithmic  system,  265. 
Moment  of  a  force,  288. 
Multiple  points,  253. 
Multiplication,  abridged,  356. 
Multiplication  of  determinants,  84. 

Napierian  logarithms,  264. 
Natural  logarithms,  264. 
Negative  roots,  166. 
Newton's  method  of  approximation, 

162. 
Nodal  line,  355. 
Node,  252. 
Non-linear     equations     representing 

lines,  68. 
Normal  form,  61,  296. 
Normal  to  ellipse,  208  ;    to  parabola, 

181,  182  ;   to  any  surface,  349. 
Numerical  equations,  158-168. 

Oblate,  334. 

Oblique  axes,  6,  7,  38,  278. 
Octant,  277. 
Ordinary  point,  251. 
Ordinate,  5. 
Origin,  1,  4,  277. 

Orthogonal  substitution,  352  ;  trans- 
formation, 352. 

Parabola,  131-142,  169-197,  229, 
244-245;  Cartesian  equation,  171 ; 
polar  equation,  169-170 ;  referred 
to  diameter  and  tangent,    190. 

Paraboloid,  elliptic,  340 ;  hyper- 
bolic, 340 ;   of  revolution,  341. 

Parallel,  335  ;   circle,  335. 

Parallelism,  28,  33,  59,  285. 

Parallelogram  law,  19,  120. 

Parameter,  107,  109 ;  equations  of 
circle,  109;  of  ellipse,  220;  of 
hyperbola,  220 ;   of  parabola,  189. 

Pascal's  triangle,  154. 

Peaucellier's  cell,  109. 

Pencil  of  circles,  107 ;  of  lines,  67 ; 
of  parallels,  67 ;  of  planes,  303  ;  of 
spheres,  329. 

Pendulum,  134, 


368 


INDEX 


Permutations,  70-73. 

Perpendicularity,  28,  33,  59,  285. 

Phase,  124. 

Plane,  292-306  ;  through  three  points, 
295. 

Plotting  by  points,  131. 

Points  of  inflection,  144. 

Polar,  102,  104,  326  ;  angle,  16  ;  axis, 
16 ;  coordinates,  16,  290 ;  equa- 
tion of  circle,  91  ;  of  conic,  224-225; 
of  line,  60 ;  of  parabola,  169-170 ; 
representation  of  complex  num- 
bers, 124. 

Pole,  16. 

Pole  and  polar,  102,  104,  326. 

Poles,  334. 

Polynomial,  148-157  ;  curve,  155-157. 

Power  of  a  point,  106,  328. 

Principal  diagonal,  47. 

Projectile,  135,  142. 

Projecting  cylinders,  321 ;  planes 
of  a  line,  309-311. 

Projection,  18-21,  280-281,  284. 

Prolate,  334. 

Proportional  quantities,  24. 

Pulleys,  27,  31,  38,  268. 

Pythagorean  relation,  282. 

Quadrant,  5. 

Quadratic    equation,    92 ;     function, 

131-142. 
Quadric  surfaces,  332-^50,  342. 

Radical  axis,  106,  328,  329 ;  center, 
107,  328,  329 ;  plane,  328. 

Radius  vector,  16,  282,  290. 

Rate  of  change,  29,  149 ;  of  interest, 
29,  35. 

Rational  numbers.  111. 

Real  axis,  117;  numbers,  113;  roots, 
160-167. 

Reciprocal  polars,  327. 

Rectangular  coordinates,  6 ;  hyper- 
bola, 203. 

Reduction  to  normal  form,  62,  297. 

Regula  falsi,  161. 

Related  quantities,  14. 

Remainder  theorem,  163. 

Removal  of  term  in  xy,  238. 

Resultant,  19,  280. 

Right-handed  trihedral,  354. 


Rotation  of  axes,  235-236;  of  co- 
ordinate trihedral,  352-355. 

Row,  41,  47. 

Rule  of  false  position,  161. 

Ruled  surfaces,  347-349. 

Rulings  on  hyperboloid  of  one  sheet, 
348  ;  on  hyperbolic  paraboloid,  349. 

Second  derivative,  144. 

Secondary  diagonal,  47. 

Sheaf  of  planes,  304. 

Shearing  force,  196-197. 

Shortest  distance  of  two  lines,  313- 
314. 

Simple  point,  251. 

Simpson's  rule,  193. 

Simultaneous  linear  equations,  39- 
48,  81-83,  302. 

Simultaneous  linear  and  quadratic 
equations,  94. 

Sine  curve,  261. 

Skew  symmetric  determinant,  84. 

Slope,  24  ;  of  ellipse,  207  ;  of  hyper- 
bola, 210;  of  parabola,  139-140, 
176  ;   of  secant  of  parabola,  138. 

Slope  form  of  equation  of  line,  26. 

Sphere,  317-331 ;  through  four  points, 
319. 

Spherical  coordinates,  290. 

Spheroid,  334. 

Spinode,  252. 

Spiral  of  Archimedes,  259. 

Square  root  of  complex  number,  129. 

Statistics,  14. 

Straight  line,  23. 

Strophoid,  260. 

Subnormal  to  parabola,  181. 

Substitutions,  270. 

Subtangent  to  parabola,  180. 

Sum  of  two  determinants,  52,  78. 

Superposable  trihedrals,  354. 

Surface,  292  ;   of  revolution,  335-336. 

Suspension  bridge,  188. 

Symmetric  determinant,  84. 

Symmetry,  136-138,  215. 

Synthetic  division,  164. 

Tangent  to  algebraic  curve  at  origin, 
250-253;  to  circle,  97;  to  ellipse, 
206,  213;  to  hyperbola,  210;  to 
parabola,  139,  180,  182. 


INDEX 


369 


Tangent  cone  to  sphere,  324. 

Tangent  curve,  261. 

Tangent  plane  to  ellipsoid,  346 ;  to 

hyperboloids,  347 ;  to  paraboloids, 

347 ;    to  quadric  surfaces,  347 ;  to 

sphere,  322. 
Taylor's  theorem,  168. 
Temperature,  15,  31,  270. 
Tetrahedron  volume,  301. 
Thermometer,  2,  31,  35. 
Transcendental  curves,  262. 
Transformation     from    cartesian    to 

polar    coordinates,     16,     290-291; 

to   center,    226,    240;     to   parallel 

axes,  12,  239. 
Translation  of  axes,  12,  233-235 ;  of 

coordinate  trihedral,  287. 


I  Transposition,  50,  78. 
Transverse  axis,  203. 
Trochoid,  258. 

Uniform  motion,  30,  69. 
Units,  5. 

Vector,  18,  119,  280. 
Vectorial  angle,  16. 
Velocity,  30,  31. 
Versiera,  256. 

Vertex  of  parabola,  132,  170. 
Vertices  of  ellipse,    198 ;    of  hyper- 
bola, 202. 
Volume  of  tetrahedron,  301. 

Water  gauge,  2. 
Whispering  galleries,  212. 


^T^HE   following  pages    contain   advertisements  of  a 
few  of  the   Macmillan  books  on  kindred  subjects. 


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TRIGONOMETRY 

BY 

ALFRED   MONROE   KENYON 

Professor  of  Mathematics,  Purdue  University 

And  LOUIS   INGOLD 

Assistant  Professor  of  Mathematics,  the  University  of 
Missouri 

Edited  by  Earle  Raymond  Hedrick 

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FROM  THE  PREFACE 

The  book  contains  a  minimum  of  purely  theoretical  matter.  Its  entire 
organization  is  intended  to  give  a  clear  view  of  the  meaning  and  the  imme- 
diate usefulness  of  Trigonometry.  The  proofs,  however,  are  in  a  form  that 
will  not  require  essential  revision  in  the  courses  that  follow.  .  .  . 

The  number  of  exercises  is  very  large,  and  the  traditional  monotony  is 
broken  by  illustrations  from  a  variety  of  topics.  Here,  as  well  as  in  the  text, 
the  attempt  is  often  made  to  lead  the  student  to  think  for  himself  by  giving 
suggestions  rather  than  completed  solutions  or  demonstrations. 

The  text  proper  is  short;  what  is  there  gained  in  space  is  used  to  make  the 
tables  very  complete  and  usable.  Attention  is  called  particularly  to  the  com- 
plete and  handily  arranged  table  of  squares,  square  roots,  cubes,  etc.;  by  its 
use  the  Pythagorean  theorem  and  the  Cosine  Law  become  practicable  for 
actual  computation.  The  use  of  the  slide  rule  and  of  four-place  tables  is 
encouraged  for  problems  that  do  not  demand  extreme  accuracy. 

Only  a  few  fundamental  definitions  and  relations  in  Trigonometry  need  be 
memorized;  these  are  here  emphasized.  The  great  body  of  principles  and 
processes  depends  upon  these  fundamentals;  these  are  presented  in  this  book, 
as  they  should  be  retained,  rather  by  emphasizing  and  dwelling  upon  that 
dependence.  Otherwise,  the  subject  can  have  no  real  educational  value,  nor 
indeed  any  permanent  practical  value. 


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THE  CALCULUS 

BY 

ELLERY  WILLIAMS  DAVIS 

Professor  of  Mathematics,  the  University  of  Nebraska 

Assisted   by  William  Charles   Brenke,  Associate   Professor  of 
Mathematics,  the  University  of  Nebraska 

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as  it  is  possible  to  do  without  venturing  into  technical  fields  whose  subject 
matter  is  itself  unknown  and  incomprehensible  to  the  student,  and  without 
abandoning  an  orderly  presentation  of  fundamental  principles. 

The  same  general  tendency  has  led  to  the  treatment  of  topics  with  a  view 
toward  bringing  out  their  essential  usefulness.  Rigorous  forms  of  demonstra- 
tion are  not  insisted  upon,  especially  where  the  precisely  rigorous  proofs 
would  be  beyond  the  present  grasp  of  the  student.  Rather  the  stress  is  laid 
upon  the  student's  certain  comprehension  of  that  which  is  done,  and  his  con- 
viction that  the  results  obtained  are  both  reasonable  and  useful.  At  the 
same  time,  an  effort  has  been  made  to  avoid  those  grosser  errors  and  actual 
misstatements  of  fact  which  have  often  offended  the  teacher  in  texts  otherwise 
attractive  and  teachable. 

Purely  destructive  criticism  and  abandonment  of  coherent  arrangement 
are  just  as  dangerous  as  ultra-conservatism.  This  book  attempts  to  preserve 
the  essential  features  of  the  Calculus,  to  give  the  student  a  thorough  training 
in  mathematical  reasoning,  to  create  in  him  a  sure  mathematical  imagination, 
and  to  meet  fairly  the  reasonable  demand  for  enlivening  and  enriching  the 
subject  through  applications  at  the  expense  of  purely  formal  work  that  con- 
tains no  essential  principle. 


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GEOMETRY 

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WALTER   BURTON   FORD 

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And  CHARLES   AMMERMAN 

The  William  McKinley  High  School,  St.  Louis 

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in  the  University  of  Missouri 

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II.  As  treated  in  this  book,  geometry  functions  in  the  thought  of  the 
pupil.     It  means  something  because  its  practical  applications  are  shown. 

III.  The  logical  as  well  as  the  practical  side  of  the  subject  is  emphasized. 

IV.  The  arrangement  of  material  is  pedagogical. 

V.  Basal  theorems  are  printed  in  black-face  type. 

VI.  The  book  conforms  to  the  recommendations  of  the  National  Com- 
mittee on  the  Teaching  of  Geometry. 

VII.  Typography  and  binding  are  excellent.  The  latter  is  the  reenforced 
tape  binding  that  is  characteristic  of  Macmillan  textbooks. 

"  Geometry  is  likely  to  remain  primarily  a  cultural,  rather  than  an  informa- 
tion subject,"  say  the  authors  in  the  preface.  "  But  the  intimate  connection 
of  geometry  with  human  activities  is  evident  upon  every  hand,  and  constitutes 
fully  as  much  an  integral  part  of  the  subject  as  does  its  older  logical  and 
scholastic  aspect."  This  connection  with  human  activities,  this  application 
of  geometry  to  real  human  needs,  is  emphasized  in  a  great  variety  of  problems 
and  constructions,  so  that  theory  and  application  are  inseparably  connected 
throughout  the  book. 

These  illustrations  and  the  many  others  contained  in  the  book  will  be  seen 
to  cover  a  wider  range  than  is  usual,  even  in  books  that  emphasize  practical 
applications  to  a  questionable  extent.  This  results  in  a  better  appreciation 
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truer  conception  of  the  wide  scope  of  its  application. 

The  logical  as  well  as  the  practical  side  of  the  subject  is  emphasized. 

Definitions,  arrangement,  and  method  of  treatment  are  logical.  The  defi- 
nitions are  particularly  simple,  clear,  and  accurate.  The  traditional  manner 
of  presentation  in  a  logical  system  is  preserved,  with  due  regard  for  practical 
applications.     Proofs,  both  forraal  and  informal,  are  strictly  logical. 


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Elements  of  Theoretical  Mechanics 

BY 

ALEXANDER   ZIWET 

Cloth,  8vo,  404  pp.,  $4.00  net 

The  work  is  not  a  treatise  on  applied  mechanics,  the  applications  being 
merely  used  to  illustrate  the  general  principles  and  to  give  the  student  an  idea 
of  the  uses  to  which  mechanics  can  be  put.  It  is  intended  to  furnish  a  safe 
and  sufficient  basis,  on  the  one  hand,  for  the  more  advanced  study  of  the  sci- 
ence, on  the  other,  for  the  study  of  its  more  simple  applications.  The  book 
will  in  particular  stimulate  the  study  of  theoretical  mechanics  in  engineering 
schools. 


Introduction  to  Analytical  Mechanics 

BY 

ALEXANDER   ZIWET 

Professor  of  Mathematics  in  the  University  of  Michigan 

And  peter   FIELD,  PH.D. 

Assistant  Professor  of  Mathematics  in  the  University  of 
Michigan 

Cloth,  j2mo,  378  pp.,  $1.60  net 

The  present  volume  is  intended  as  a  brief  introduction  to  mechanics  for 
junior  and  senior  students  in  colleges  and  universities.  It  is  based  to  a  large 
extent  on  Ziwet's  "Theoretical  Mechanics  ";  but  the  applications  to  engineer- 
ing are  omitted,  and  the  analytical  treatment  has  been  broadened.  No  knowl- 
edge of  differential  equations  is  presupposed,  the  treatment  of  the  occurring 
equations  being  fully  explained.  It  is  believed  that  the  book  can  readily  be 
covered  in  a  three-hour  course  extending  throughout  a  year.  The  book  has, 
however,  been  arranged  so  that  certain  omissions  may  be  easily  made  in  order 
to  adapt  the  book  for  use  in  a  shorter  course. 

While  more  prominence  has  been  given  to  the  analytical  side  of  the  sub- 
ject, the  more  intuitive  geometrical  ideas  are  generally  made  to  precede  the 
analysis.  In  doing  this  the  idea  of  the  vector  is  freely  used;  but  it  has 
seemed  best  to  avoid  the  special  methods  and  notations  of  vector  analysis. 

That  material  has  been  selected  which  will  be  not  only  useful  to  the  begin- 
ning student  of  mathematics  and  physical  science,  but  which  will  also  give  the 
reader  a  general  view  of  the  science  of  mechanics  as  a  whole  and  afford  him 
a  foundation  broad  enough  to  facilitate  further  study. 


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